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First Memoir On some Properties of the Perfect Positive Quadratic Forms Georges Fedosevich Voronoi Monograph Translations Series Series Editor: Vaen Sryayudhya Translated by Kit Tyabandha, Ph.D. God’s Ayudhya’s Defence Bangkok 7 th February, 2007 Catalogue in Publication Data G. F. Voronoi On some Properties of the Perfect Positive Quadratic Formes: - Bangkok, Kittix, GAD, 2007 116 p. 1. On some Properties of the Perfect Positive Quadratic Formes. I. Voronoi, G. F. II. Tyabandha, Kit III. Mathematics. 510 ISBN 978-974-88248-9-5 Translation © Kit Tyabandha, 2007 All rights reserved Published by God’s Ayudhya’s Defence A Non-profit School of Mathematics, Language and Philosophy 1564/11 Prajarastrasa'i 1 Road Bangzue, Bangkok 10800, Thailand Editor Vaen Sryayudhya Translator Kit Tyabandha, Ph.D. Typeset using Tf^X God’s Ayudhya’s Defence is the only trademark relevant to the publication of this book All other trademarks and trade names are mentioned solely for explanation In Thailand, Baht 1,200 Elsewhere consult distributor’s quotation Distribution Internet-searchable world-wide To God G. F. Voronoi (1908) Monograph Translations Series Preface George Fedosevich Voronoi was born on 28 th April 1868 , in Zhuravka, Poltava Guberniya, Russia, which is now in Ukraine. He died on 20 th November 1908 in Warsaw, Poland. Both his master’s degree, 1894, on the algebraic integers associated with the roots of an irre¬ ducible cubic equation and his doctoral thesis on algo¬ rithms for continued fractions were awarded the Bun- yakovsky prize by the St. Petersburg Academy of Sci¬ ences. But he decided that he wanted to teach at the Warsaw University where he extended work by Zolotarev on algebraic numbers and the geometry of numbers. He met Minkowski in 1904 at an international conference in Heidelberg. The three papers by Voronoi all appeared in the same journal, the influential and prestigious Journal filr die reine und angewandte Mathematik [Journal for the pure and applied mathematics]. This journal was a leading mathematical journal during 19 th and early 20 th cen¬ turies when most of its publications appeared either in French or in German. In French it is called Journal de Crelle or Crelle’s Journal after the name of its founder in 1826 by August Leopold Crelle (1780—1855). Orig¬ inally Crelle intended the journal to emphasise equally both pure and applied mathematics. But the policy soon changed and it has been dealing solely with pure mathe¬ matics since the start of his second and short-lived Jour¬ nal fur die Bankunst (1829—1851) to deal the application side. The main idea is quite simple. It is that space can be partitioned into a set of regions, each surrounding a single point which is sometimes called a nucleus. Every point in a given region is closer to its own nucleus than to any nuclei of other regions. This idea has found so many applications in nature that I think it is as beau¬ tiful as the golden ratio 1+ f^ is. It has been used to study the forest fire and it has been used to study the structure of the distribution of galaxy. In fact the inter- God’s Ayudhya’s Defence 7 th February, 2007 i Monograph Translations Series G. F. Voronoi (1908) nal structure of many things has proved to be Voronoi, things in nature as well as man-made ones, for example plant cells and filtering membranes. Looking at a two- dimensional Voronoi structure will remind one of plant or animal cells partitioned by straight walls which re¬ sult from cosiness of circular cells growing among their neighbours. Voronoi’s lifelong work was in theory of numbers and is divided into three groups, namely algebraic theory of numbers, analytic theory of numbers, and geometry of numbers. His three papers translated here make up two of the planned series of memoirs to apply the principal of Continuous Hermite (Charles Hermite, 1822—1901) pa¬ rameters to problems of the arithmetical theory of def¬ inite and indefinite quadratic forms. He had completed only two of the series when he died in Warsaw, Poland on 20 th November 1908 , before the last one appeared in print in 1909. A short obituary written by Kurt Hensel (1861 — 1941) was included at the end of the paper, which is also included here in translation. The first paper gives characteristics of complete qua¬ dratic forms. In it Voronoi solved the question posed by Hermite on the upper limit of the minima of the posi¬ tive quadratic forms for a given discriminant of n vari¬ ables. Zolotarev (Egor Ivanovich Zolotarev, 1847—1878) and Korkin (Aleksandr Nikolaevich Korkin) had given solutions for n = 4 and n = 5. Voronoi gave an algo¬ rithmic solution for any n. He did this with the help of the methods of the geometrical theory of numbers. The present volume gives an English translation of this paper. The second and the third papers deal with simple parallelepipeds, that is polyhedra with parallelograms as all their faces. He gave the determination of all possi¬ ble methods of filling an n-dimensional Euclidean space with identical convex non-intersecting polyhedra (par¬ allelepipeds) which have completely contiguous bound¬ aries. A solution of this problem for three-dimensional space had been given by Fedorov (Evgraf Stepanovich Fyodorov, 1853—1919) who was a crystallographer al¬ though the proof he gave is said to be incomplete. Min- li 7 th February, 2007 God’s Ayudhya’s Defence G. F. Voronoi (1908) Monograph Translations Series kowski (Hermann Minkowski, 1864—1909) showed in 1896 that the parallelepipeds must have centres of symmetry. He also demonstrated that the number of their bound¬ aries did not exceed 2(2 n — 1). Voronoi imposed further the requirement that n + 1 parallelepipeds converge at each summit and solved the problem for these conditions completely. Voronoi’s collected works appeared in three volumes under the title of Sobranie sochineny, Kiev, 1952—1953. He is sometimes referred to as belonging to the St. Pe¬ tersburg school of the Theory of Numbers. This must have been the Petersburg Mathematical School some¬ times called Chebyshev School or Petersburg School. It was founded by Chebyshev (Pafnuty Lvovich Chebyshev, 1821—1894) and had prominent figures as Grave (Dmitri Aleksandrovich Grave), Krylov (Aleksei Nikolaevich Kr¬ ylov, 1863—1945), Lyapunov (Aleksandr Mikhailovich L- yapunov, 1857—1918), Markov (Andrei Andreyevich Mar¬ kov, 1856—1922), Sohotski (Yulian-Karl Vasilievich So- khotsky, 1842—1927), Steklov (Vladimir Andreevich Ste- klov, 1864—1926), Korkin, K. A. Posse, and A. V. Vas- siliev. iii God’s Ayudhya’s Defence 7 th February, 2007 Monograph Translations Series G. F. Voronoi (1908) iv 7 th February, 2007 God’s Ayudhya’s Defence G. F. Voronoi (1908) Monograph Translations Series New applications of continuous parameters to the theory of the quadratic form First Memoir On some properties of the perfect positive quadratic forms by Mr. Georges Voronoi in Warsaw [Journal fur die reine und angewandte Mathematik] [V. 133, p. 97-178, 1908] [translated by K N Tiyapan] Introduction Hermite had introduced in the theory of numbers a new and fruitful principle, namely: being given a set ( x ) of systems (x\ , X 2 , ■ ■ ■ , x n ) for all the values of xi,X 2 ,---> x n , one associates with the set ( x ) a set ( R ) composed of the domains in a manner such that by studying the set (i?) one studies at the same time the set ( x ) . Hermite has shown f numerous applications of the f Hermite. Extraits de lettres de M. Ch. Hermite a M. Jacobi sur differents objets de la theorie des nombres. [Excerpts from letters of Mr. Ch. Hermite to Mr. Ja¬ cobi on various subjects in the theory of numbers] (This Journal V. 40, p. 261) Hermite. Sur l’lntroduction des variables continues dans la theorie des nombres. [On the introduction of the con¬ tinuous variables in the theory of numbers] (This Journal V. 41, p. 191) Hermite. Sur la theorie des formes quadratiques. [On the theory of quadratic forms] (This Journal V. 47, p 313) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 5 Monograph Translations Series G. F. Voronoi (1908) new principle to the generalisation of continuous frac¬ tions, to the study of algebraic units, etc. The ideas of Hermite have been developed in the works of Mr.’s Zolotareff, Charve, Selling, Minkowski. X I intend to publish a series of Memoires in which I shall show new applications of the principle of Hermite to the various problems of the arithmetic theory of def¬ inite and indefinite quadratic forms. In this Memoire, I study the properties of the min¬ imum of positive quadratic forms and of their various representations by systems of integers. Hermite has discovered an important property of the minimum M of positive quadratic forms J2 a ij x i x j i n n variables and of the determinant D, namely: n-1 M<(f) - VD, and he has demonstrated numerous applications of this $ Zolotareff. On an indeterminate equation of the third degree (Petersbourg, 1869, in Russian.) Zolotareff. Theory of complex integers with applications to the integral calculus. (Petersbourg, 1874, in Russian.) Charve. De la reduction des formes quadratiques ter- naires positives et de leur application aux irrationelles de troisieme degre. [Of the reduction of positive ternary quadratic forms and of their application to the irration¬ als of third degree] (Suppl. to V. IX of Annales Scien- tifiques de l’Ecole Normale Superieure, 1880) Selling. Uber die binaren und ternaren quadratischen Formen. [On the binary and ternary quadratic forms] (This Journal, V. 77, p. 143) Minkowski. Geometrie der Zahlen. [Geometry of num¬ bers] (Leipzig, 1896) 6 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) formula. Monograph Translations Series In a letter to Jacobi, Hermite has said §: “That which precedes sufficiently indicates an infin¬ ity of other analogous consequences which, all, will de¬ pend on the difficult study of an exact limit of the min¬ imum of any definite form. Thereupon I then form only one conjecture. My first studies in the case of a form in n variables of the determinant D have given me the limit re — 1 (f)^~ Vd, I am inclined to presume, but without being re -1 able to demonstrate that the numerical coefficient (|) 2 has to be replaced by y== ” Mr.’s Korkine and Zolotareff has under taken the study of the exact limit of the minimum of positive quad¬ ratic forms of the same determinant. By indicating with M(aij ) the minimum and with D ( dij ) the determinant of the form Y&ijXiXj , one will have the minimum M(aij ) M (cLjj ) y/D(dij ) of a positive quadratic form with determinant 1. By virtue of the theorem of Hermite the function M(aij) verifies the inequality < (|)^« § This Journal. V 40, p. 296 f Mr. Minkowski has demonstrated an upper limit of the function M(aij) M(aij) < n much simpler than that from Hermite. ( Minkowski. Uber die positiven quadratischen For- men und uber kettenbruchahnliche Algorithmen. [On the positive quadratic forms and on continued fraction algorithm] This Journal V. 107, p. 291) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 7 Monograph Translations Series G. F. Voronoi (1908) therefore it is bounded within the set (/) of all the pos¬ itive quadratic forms of real coefficients. Mr.’s Korkine and Zolotareff have demonstrated f that the function _M(aqj) possesses many maxima in the set (/) which correspond to the various classes of equiv¬ alent positive quadratic forms. The limit y^ l+1 indicated by Hermite in the letter to Jacobi (source cited) is only a maximum value of the function M(aij). The binary and ternary positive quadratic forms pos¬ sess a single maximum which is therefore, in this case, the exact limit of values of the function M(aij). Reckoning from the number of variables n > 4, one meets many maxima of the function Af(ajj). Mr.’s Korkine and Zolotareff have found many values of various maxima of the function M(oqj) which exceed the limit 2 . „ indicated by Hermite, but do not exceed the limit 2. The study of the exact limit of the minimum of pos¬ itive quadratic forms of the equal determinant comes down, after Mr.’s Korkine and Zolotareff, to the study of all the various classes of positive quadratic forms to which correspond the maximum values of the function M(aij). The maximum maximorum of values of the function M(aij ) is the largest value of the function M(aij) which presents a numerical function as p,(n). Mr.’s Korkine and Zolotareff have determined the following values of the function p,(n): M2) = y|, M3) = #2, M4) = M5) = VE, • f Korkine and Zolotareff. Sur les formes quadratiques. [On the quadratic forms] Mathematische Annalen, V. VI, p. 366 and V. XI, p. 242 8 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series They have called extreme the quadratic forms which yield to the function Af(a^) a maximum value. The extreme quadratic forms enjoy an important pr¬ operty, namely: I. Any extreme quadratic form is determined by the value of its minimum and by all the representations of the minimum. Mr.’s Korkine and Zolotareff have determined all the classes of extreme forms in 2, 3, 4 and 5 vertices. By studying these extreme forms, I have observed that they are all well defined by the property (I). There is only reckoning from positive forms in six variable wh¬ ich I have encountered positive quadratic forms which enjoyed the property (I) and are not of extreme forms. I call “perfect” any positive quadratic form which enjoys the property (I). I demonstrate that the set of all the perfect forms in n variables can be divided into classes the number of which is finite. All extreme form being, by virtue of the property I, a perfect form, it results in that the function ju(n) presents the maximum of values of the function M(aij) which correspond to the various classes of perfect forms. I have established an algorithm for the search of var¬ ious perfect forms by introducing a definition of contigu¬ ous perfect forms. To that effect, I make correspond to the set (<p) of all the perfect forms in n variables a set (f?) of domains in n bH~ 1 ) dimensions determined with the help of linear inequalities. The set (f?) of domains in n ( n + 1 '> dimensions presents a partition of the set (/) of all the positive quadratic God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 9 Monograph Translations Series G. F. Voronoi (1908) forms in n variables. Each domain R possesses in the set ( R ) a contiguous domain which is well determined by any one face in - n ( n + 1 ') — i dimensions of the domain R. I demonstrate that the domain R corresponding to the perfect form tp(x±, x 2 , ■ ■ ■ , x n ) being determined by the linear inequalities ^Pifaij >0, (fe = 1,2, . . ., ct) one will have a perfect forms defined by the equalities <Pk(xi, X2, ■ ■ ■ , X n ) = (p(x!, X 2 , ■ ■ ■ , X n ) + Pk^k(xi, X 2 , ■ ■ ■ , X n ), (k= 1,2,...,a) (1) where ^k(xi,x 2 , ■ ■ ■ ,x n ) = 'E'P^XiXj, provided that the positive parameter p *. (k = 1,2, . . ., a) presents the smallest value of the function <t>(xi,x 2 , . . . , x n ) - M -<P(x 1 ,x 2 , • • • , x n ) where (xi, x 2 , . . . , x n ) < 0 and M is minimum of the form 4>(xi,x 2 , . . . , x n ). I call “contiguous to the perfect form <fi(xi, x 2 , . . . , x n )” the perfect forms (1). Any substitution in integer coefficients and with de¬ terminant ±1 belonging to the group g of substitutions which do not change the form cf> permute only the forms (1). One can, therefore, divide the forms (1) into classes of equivalent forms with the help of substitutions of the group g. By choosing one form in each class, one will have a system of perfect forms contiguous to the perfect form (p which can replace the system (1). By proceeding in this manner, one can obtain a sys¬ tem complete of representatives of various classes of per¬ fect forms. 10 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series The corresponding domains will form complete sys¬ tem of representatives of various classes of the set ( R ). I have remarked that a similar system R, Ri, f?2, • • • J fir-1 (2) of domains of the set (f?) can serve towards the reduction of positive quadratic forms. I call reduced any positive quadratic form belonging to one of the domains (2). It results from this definition: I. Any positive quadratic form can be transformed into an equivalent reduced form, with the help of a substitution which presents a product of substitutions belonging to a series of substitutions Si , S 2 , , S m which depend only on the choice of the system (2). II. Two reduced forms can be equivalent only provided that the corresponding substitution belonged to a series of substitutions the number of which is finite. The weak point of the new method of reduction of positive quadratic forms, demonstrated in this Memoire, consists in that the number of substitutions which trans¬ form into itself the domains of the set (f?) or their faces is, in general, very large. The application of the general theory demonstrated in this Memoire to the numerical examples will be partic¬ ularly facilitated if one knew how to solve the following problem: Being given a group G of substitutions which trans¬ form into itself a domain R, one would like to partition this domain into equivalent parts the number of which will be equal to the number of substitutions of the group G and God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 11 Monograph Translations Series G. F. Voronoi (1908) on condition that the number of faces in n ( n +D — 1 dimen¬ sions of domains obtained be the smallest possible. I show in this Memoire the solution of the problem introduced in two cases: n = 2 and n = 3. From the number of variable n > 4, I do not know any practical solution of the problem posed. First Part General theory of perfect positive quadratic forms and domains which correspond to them Definition of perfect quadratic forms. Let <t>(xi, x 2 , ■ ■ ■ , x n ) = y ^ajjXjXj (1) be any positive quadratic form. By indicating with Oil) 1 21) • • • ) Ini') i (l 12, 122, • • • On 2 ), ■ ■ ■ , (l Is, hs, ■ ■ ■ , Ins) (2) the various representations of the minimum M of the form J2aijXiXj, one will have the equalities ^ aijUkljk = M, (k = 1,2, . . ., s) (3) One will not consider in the following the two sys¬ tems (Ilk, l2k, • • • ) Ink) and ( Ilk, l2k, • • • , Ink), (k= 1,2 as different and one will arbitrarily choose one of these systems. 12 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series On the ground of the supposition made, one will have the inequality ^2 CLijXiXj > M provided that a system (xi, X 2 , ■ ■ ■, x n ) of integer values of variables x\, x?, . . . , x n did not belong to the series (2), excluding the system x\ = 0, X 2 = 0, . . . , x n = 0. By considering the equalities (3) as the equations which serve to determine n ^ n 2 +1 ' 1 coefficients of the quad¬ ratic form 'ZaijXiXj, one will have only two cases to ex¬ amine: 1. ) there exist a finite number of solutions of equa¬ tions (3), 2. ) the equations (3) admit only a single system of solutions. Let us examine the first case, Let us suppose that there exists an infinite number of solutions of equations (3). One will find in this case an infinite number of values of parameters Pij = Pjii ( ^ = 1 j 2, . . . , n ; j = 1,2,...,77.) verifying the equations ^ ' Pijlikljk ^ 0, (fc = 1, 2, . . . , s) (^) the values p^j = 0, i = 1,2, . . ., n; j = 1,2, . . ., n being ex¬ cluded. By indicating '&(x 1 ,X 2 , ■ ■ ■ ,X n ) = ^PijXiXj , let us consider the set of positive quadratic forms deter¬ mined by the equality f(x 1 ,X 2 ,---,X n ) = p(x 1 ,X 2 ,---,Xn) + P^(x 1 ,X 2 ,---,X n ), (5) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 13 Monograph Translations Series G. F. Voronoi (1908) the parameter p being arbitrary. For a quadratic form determined by the equality (5) to be positive, it is necessary and sufficient that the cor¬ responding value of the parameter p be continuous in a certain interval -R! < p < R. It can turn out that R = +oo, in this case the lower limit —R 1 will be finite. By replacing in the equality (5) the form A/ (x±, x 2 , . . . , x n ) by the form — SE^xi, X 2 , . . . , x n ), that which is permitted by virtue of (4), one will have the interval -R < p < R 1 , therefore one can suppose that the upper limit R is fi¬ nite. The corresponding quadratic form, determined by the equality f(x!,x 2 , . . . ,x n ) = (p(x 1} X 2 , . . . ,x n ) + RA?(x 1,052, . . . , X n ) , will not be positive, but it will not have negative values either; one concludes that least for a system (£i,£ 2 , . .., £ n ) of real values of variables x\, x 2 , ■ ■ . , x n the form /(x i, x 2 , . . • ,x n ) attains in its value the smallest which is zero, and it follows that the system (£i, £ 2 , ■ ■ ■ ; £ n ) verifies the equation df_ d£,i = 0. (i = 1,2, . . ., n) By eliminating from these equations £ 1 , £ 2 , ■ ■ ■, €n one obtains the equation an + -R-Pn, ai 2 + RP 12 , ■ ■ • j ^ln “h RP±n D(R ) = a 2 i + RP 21 , a> 22 + RP 22 , ■ ■ • j a 2n + RP2n = 0 a n i “l - RRn i> ®n2 “1“ RPn2i • • • j Q'nn “1“ RPnn The smallest positive root of this equation presents the value of R searched for. 14 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series Let us examine the set (/) of positive quadratic forms determined by the equality (5) with condition 0 < p < R. ( 6 ) Theorem. To the set (/) belongs a quadratic form pi(x±, X 2 , ■ ■ ■ ,x n ) which is well determined by the following con¬ ditions: 1. all the representations of the minimum M of the form p(xi, X 2 , ■ ■ ■, x n ) are also representations of the min¬ imum M of the form <pi(xi, x ^, • • • , x n ), 2. the form cpi(xi, X 2 , ■ ■ ■, x n ) moreover possesses at least another representation of the minimum M. Let us indicate by M(p) the minimum and by D(p) the determinant of the quadratic form f(x 1 , 2 : 2 , defined by the equality (5) with condition (6). By virtue fo the theorem by Hermite, one will have the inequality M(p)<p(n)tfD(pj. (7) We have demonstrated that D(R ) = 0, it results in that a value of the parameter p can be chosen in the interval (6) such that the inequality p(n) V-D(p) < M holds. One will have, because of (7), M (p) < M. (8) Let us indicate by (li, I 2 , ■ ■ ■ , l n ) a representation of the minimum M(p) of the form f (x 1 , X 2 , ■ ■ ■ , x n ) verifying the inequality (8). God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 15 Monograph Translations Series G. F. Voronoi (1908) One will have I2, ■ ■ ■ , In) + P'S' Ol, I2, • • • , In) < M, ( 9 ) and as a result <p(h,h, ■ ■ ■ ,l n ) > M and d>(Zi, l 2 , ■ ■ ■, l n ) < 0 . (10) This posed, let us find the smallest value of the func¬ tion <p(xi, x 2 , ■ ■ ■, x n ) - M -^/(x 1 ,x 2 , • • • , X n ) determined with condition ^(zi, x 2 , ■ ■ ■ , x n ) < 0. ( 11 ) ( 12 ) To that effect, let us examine the inequality V(xi,x 2 , • ■ ■ , x n ) - M < (p(h,l 2 , ■ ■ ■ , l n ) ~ M ^ (x 1, X 2 , ... , X n ) *(h,l 2 ,...,ln) By virtue of (9), (10) and (12), one will have <p(x 1} x 2 , . . . ,x n ) + p'&(x 1 ,x 2 , . . . ,x n ) < M. The quadratic form p(x\,x 2 , ■ ■ ■ , x n ) + pd/( x±,x 2 , . . . , x n ) being positive, there exists only a limited number of integer values of x±,x 2 ,...,x n verifying this inequality. Among these systems are found all the systems which give back to the function (11) the smallest value deter¬ mined with condition (12). Let us indicate by (it it 11 \ (ltt 111 lit \ (](r) j(r ) /(O) ri) ) 4 n / ’ \ l l’ • • • ) ’ ’ • • ) r 1 ?‘ , 2 > • • • ? 1 n ) all the representations of the positive minimum pi of the function (11). By declaring <Pl(xi, X 2 , . . . ,X n ) = p(xi,x 2 , . . . ,X n ) + pi'fy (xi, x 2 , . . . ,x n ), 16 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series one obtains the positive quadratic form ip±(xi, x 2 , • • •, x n ) the minimum M of which is represented by the systems (2) and (13), this is that which one will demonstrate without trouble. With the help of the procedure previously shown, one will determine a series of positive quadratic forms <P,<Pi,<P2, ■ ■ ■ (14) which enjoy the following property: by indicating with Sk the number of representations of the minimum of the form <Pk(k = 1,2,...), one will have the inequalities s < Si < S 2 < ■ ■ ■ (15) A similar series of positive quadratic forms of n vari¬ ables can not be extended indefinitely, this is that which we will demonstrate with the help of the following lem¬ ma. Lemma. The number of various representations of the minimum of a positive quadratic form in n variables does not exceed 2 n — 1. Let us indicate by (Zi, l 2 , ■ • •, l n ) and (l ), 1' 2 , . ■ . , l ' n ) any two representations of the minimum M of the positive quadratic form Y) a ij x i x j ■ Let us suppose that by declaring l' i = l i + 2t i , (i = 1,2, . . ,,n) (16) the number ti,t 2 ,...,t n would be integer. As ^ aijljl'j = M and aijljlj = M, by virtue of (16), it becomes ijlitj + E Gjijt it j — 0 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 17 Monograph Translations Series G. F. Voronoi (1908) One will present this equality under the form FI Q ij ~l ~ tj) (lj tj') aijtitj = aijlilj. (17) By noticing that y y Q’jji'ii'j ^ y ^ etjjiji j j one finds, by virtue of (17), F aij(li + tj)(lj + tj ) < 0, therefore it is necessary that y ' tj)) (lj tj) — 0, and consequently li + tj = 0. (i = 1, 2, . . . , n) Because of (16), one obtains l j = — li ■ (i = 1, 2n) This posed, let us divide the set (X) of all the systems (xi, X 2 , . . . , x n ) of integer values of xi,X 2 ,...,x n into 2 n classes, with regard to the modulo 2. We have demonstrated that two different representa¬ tions of the minimum M of the form ^djjXjXj will not belong to the same class; neither will any representa¬ tion of the minimum M belong to the class made up of systems (x±, X 2 , ■ ■ ■, x n ) satisfying the condition a3j=0(mod2), (i = 1, 2, . . . , n) therefore the number of various representations of the minimum of a positive quadratic form can not be greater than 2 n - 1. We have demonstrated that the series (14) of positive quadratic forms satisfying the condition (15) can not be extended indefinitely, therefore the series (14) will be terminated by a form ip/- which enjoys the following prop¬ erty: the form <is determined by the representations of its minimum. 18 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series Definition. One will call perfect any positive quadratic form which is determined by the representations of its minimum. Let us suppose that the form (1) be perfect, one will have in this case only a single system of solutions of equations (3). On the ground of the supposition made, the equa¬ tions _ ^ ' Pijlikljk — Oj (h — 1,2, . . . , s) admit only a single system of solutions Pij = Pji = 0. (i = 1,2,. . ., n; j = 1,2, . . ., n) By effecting the solution of equations (3), one ob¬ tains the equalities aij — otij AT , (i — 1,2, . . ■ , n ■ J — 1,2, . . ., n) where the coefficients ctij are rational. It results in that the perfect form is of rational coefficients. In the following one will not consider as different the perfect forms of proportional coefficients. Fundamental properties of perfect quadratic forms. Let P ( x i, x 2, . . . , n) — aijXiXj be a perfect quadratic form. Let us suppose that all the different representations of the minimum of the perfect form cp make up the series (^11)^21) • • • j lnl) i (l12i 122i ■ ■ ■ i ln2 ),...,(hs,l2s,...,l n s)- (1) By choosing any n systems in this series, let us ex¬ amine the determinant ' 'Ilf ll 2, • • • , l In > '21, 122, • • • , ^2n = ±LU. ( 2 ) n 1, l n2 j • • • , Inn God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 19 Monograph Translations Series G. F. Voronoi (1908) All the determinants that one can form this way can not cancel each other out. By supposing the contrary, one will have s equations of the form n — 1 lik = Y, li r uj k) , (i = 1,2, ... ,n-,k = 1,2, ... ,s) (3) r= 1 One will choose a system of n( ' n + 1 ' > parameters pij = Pji verifying n ( n + 1 '> equations ^ ' Pijlirljt — Oj (T — 1,2, . . . , 71 1 ] t — 1, 2, . . ., n 1) and by virtue of (3), one will have ^ ' Pijlikljk — dj (k — 1,2, • • • , £») which is impossible. The numerical value uj of the determinant (2) can not exceed a fixed limit. To demonstrate this, let us effect a transformation of the perfect form ip with the help of a substitution Xi — l ir*E r , (1 — 1, 2 , . . . , 71) r= 1 one will obtain a form ¥>'(z'i, z' 2 , • • • , x‘ n ) = Y^a’ijx’ix’j, where a’a = M. (i = 1,2, ..., n ) (4) (5) By indicating with D' the determinant of the form p>', one will have the inequality a ll a 22 ’ ’ ’ a nn — D , by virtue of the known property of positive quadratic forms. Considering (5), one obtains M n > D'. (6) 20 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series By indicating with D the determinant of the form ip, one will have, because of (2) and (4), D' = Du 2 , therefore the inequality (6) reduces to the one here: Du 2 < M n . By virtue of the theorem by Hermite, one has the inequality M < /u(n) \fl5 ; it follows that (?) Any perfect form will obviously be transformed into a form, also perfect, with the help of all linear substitu¬ tion of integer coefficients and of determinant ±1. One concludes this that there exists a finite a finite number of equivalent perfect forms. The set (p) of all the perfect forms in n variables can be divided into different classes provided that each class be made up of all the equivalent perfect forms. Theorem. The number of different classes of per¬ fect forms in n variables is finite. Let us indicate by A k — 11 k 1 1 2 k & 2 “1“ • • • “1” l nk'E n ( & — 1)2, . . . , S ) s linear forms Ai, A 2 , . . . , A s (8) f See the Memoire of Mr.’s Korkine and Zolotareff sur les formes quadratiques positives. (Mathematische An- nalen V. XI, p. 256) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 21 Monograph Translations Series G. F. Voronoi (1908) which correspond to the systems (1) of representations of the minimum of the form SI/. One establishes this way a uniform correspondence between a perfect form p and the system (8) of linear forms. Let us suppose that one had transformed the per¬ fected form p with the help of a substitution S by inte¬ ger coefficients and with determinant ±1, one will obtain an equivalent perfect form p'. Let us indicate by Ai, A' 2j ...,A' s (9) the corresponding system of linear forms. One will easily demonstrate that the substitution T, adjoint to the substitution S §, will transform the system (8) into a system (9). One concludes that a certain reduction of perfect forms can be effected with the help of the reduction of corresponding systems of linear forms. The reduction of the system (8) comes down, by virt¬ ue of (7), to the reduction of any n linear forms Ai, A 2 , • • • , A n (10) belonging to the system (8) and with determinant ±u; which does not cancel each other out. § The substitution S being defined by the equalities n Xi = ^2 ctikx'k, (i = 1, 2 , . . ., n) k =1 one calls “substitution adjoint to S” the substitution T which is determined by the equalities n ) ' OtikX k — X ^. ( i — 1 , 2 , . . . , 71 ) fc =1 22 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series One will determine with the help of the known meth¬ od a substitution T which will transform the linear forms (10) of integer coefficients into linear forms, A' 1 ,A' 2j ... J A' n (11) satisfying to the following conditions Afc = Pk,kX d - Pk + l,k x k -\-l d - . . . d - Pn,kX n , (fc=l,2,...,Tl) < PnP 22 ---Pnn = w and p kk >0, (k = 1,2,... ,n) , 0 < p k+i , k < p kk . (i = 1,2, .. . ,n - fc; fc = 1,2,...,n) The coefficients of forms (11) being integers, as a result they do not exceed fixed limits. The substitution T will transform the system (8) into a system Ai,A' 2 ,...,A' s (12) of linear forms. By examining successively the determi¬ nants of forms (Afc, A 2 , . . . , A'J, (A;, A' fc> ■ ■ ■ , A' n ), . . ., (Ai, A' 2j . . . , A' fc ), (k = n-\-l,n-\-2,...,s ) one will demonstrate that the numerical values of coef¬ ficients of all the linear forms (12) do not exceed fixed limits. The number of similar systems of linear forms in inte¬ ger coefficients being limited, it results in that the num¬ ber of different classes of perfect forms is also limited. On the domains determined with the help of linear in¬ equalities We have seen in Number 7 that the study of per¬ fect forms can be brought back to the study of certain systems of linear forms. God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 23 Monograph Translations Series G. F. Voronoi (1908) One will acquire a new basis to these studies by mak¬ ing correspond to each perfect quadratic form in n vari¬ ables a domain in n ( n + 1 '> dimensions determined with the help of linear inequalities. One will address first the general problem by study¬ ing the properties of domains determined with the help of linear inequalities. J Let us consider a system of linear inequalities PlkXl + P2kX 2 + • • • + PmkXm > 0, (k = 1,2, . . . , o) in any real coefficients. One will call point (x) any system (xi, X 2 , • • • , x m ) of real values of variables xi, X 2 , ■ ■ ■ , x m and one will indicate Vk{x ) = PlkXl + P2kX 2 + • • • + PmkXm■ ( k = 1 , 2 , . . . , (j) One will call “domain” the set R of points verifying the inequalities Uk(x) > 0. (fc= 1,2(1) Let us suppose that to the domain R belonged to points verifying the inequalities Vk(x ) >0, (fc = 1,2, . . . , ct) one will call such points interior to the domain R, and the domain R will be said to be of m dimensions. It can be the case that the domain R does not possess interior points. One will demonstrate in this case all the points belonging to the domain R verify at least one equation Vk(x ) = 0, the indice h being a value 1,2 , . . ., a. It is important to have a criteria with the help of which one could recognise whether a domain determined by the help of inequalities (1) will be in m dimensions or not. J See: Minkowski. Geometrie der Zahlen [Geometry of the numbers], No. 19, p. 39. 24 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series Fundamental principle. For a domain determined with the help of inequalities (1) to be of m dimensions, it is necessary and sufficient that the equation ^2pkVk(x) = 0 (H) k= 1 did not reduce into an identity so long as the parameters p 2 ,p 2 ,...,p a are positive or zero, the values pi = 0 , p 2 = 0, . . . , pa = 0 being excluded. The principle introduced, considered from a certain point of view, is evident, but one arrive at the rigorous demonstration of this principle only with the help of the in depth study of domains determined with the help of linear inequalities. For more simplicity, one will examine in that which follows only domains satisfying the following conditions: the equations Vk{x) = 0 (k= 1,2, ...,a) (2) can not be verified by any point, the point X\ = 0,x 2 = 0, . . ., x m = 0 being excluded. It is easy to demonstrate that the general case will always come down to the case examined. Definition. One will call edge of the domain R deter¬ mined with the help of inequalities (1) the set of points belonging to the domain R and verifying the equations y k (x) = 0, (k = 1,2, ... ,r where r < a) provided that these equations defined the values of xi,x 2 , . . . ,x m to an immediate common factor. By indicating with (£i,£ 2 , • • • ,£ m ) a point of the edge considered, one will determine all the points of the edge with the help of equalities x 1 — P^i 1 (i — 1 , 2 , . . . , n ) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 25 Monograph Translations Series G. F. Voronoi (1908) p being an arbitrary positive parameter. This results in that each edge of the domain R is well determined by any point belonging to it. Let us suppose that the domain R possesses s edges characterised by the points (£fc) = (£lfc>£ 2 fcj'-'j£mfc)' (k = 1,2, . . . , s) By declaring Xi — ^ ' Pk^ik i (i — 1 , 2 ,..., m ) ( 3 ) k = 1 where Pk> 0, (k = 1,2, . . ., s) (4) one obtains a point ( x ) belonging to the domain R, the positive or zero parameters pi,p 2 ,---,p s being arbitrary. Fundamental theorem. Let us suppose that the inequ¬ alities (1) which define the domain R satisfy the condition (*)■ The domain R will be of m dimensions and each point belonging to it will be determined by the equalities (3) with condition (4) ■ The theorem introduced is well known in the case m = 2 and m = 3. We will demonstrate that by supposing that the the¬ orem be true in the case of m — 1 variables, the theorem will again be true in the case of m variables. Let us examine first the various inequalities of the system (1). It can be the case that many among them could be put under the form S Vh(x) = P { k ] yk(x) where p { k h) > 0. k= 1 26 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series (k= 1,2,. . .,s-,p[ h) = 0) One will call such inequalities dependent and one will exclude them from the system (1). Let us suppose that the system (1) contained only independent inequalities. Their number p, on the ground of the supposition (2) made, will not be less than m. This posed, let us examine a set P k of points belong¬ ing to the domain R and verifying an equation Vh(x) = 0, (5) the indice h having a value 1,2, . . ., a. One will call “face of the domain f?” the domain P On the ground of the supposition made, the face P k will be in m — 1 dimensions. To demonstrate this, let us make correspond to any point ( x ) verifying the equation (5) a point ( u ) in m — 1 coordinates («i, U 2 , ■ ■ ■ , by declaring m — 1 Xi = ^2 a ij u j ■ (* = 1,2, . . ., m) 3 = i The system of inequalities (1) will be transformed into a system Vk(u)> 0 (k = 1,2, . . ., ct; k h) (7) of inequalities in m — 1 variables u±, u 2 , . . . , u m _ 1 . Let us suppose that one knew how to reduce the equation <7 ^2 PkVk(u) = 0 where p h = 0 and p k > 0 (k = 1, 2, . . ., a) k = 1 ( 8 ) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 27 Monograph Translations Series G. F. Voronoi (1908) into an identity. By virtue of (6), one will obtain the identity Y PkVk(x) = py h (x) where p h = 0. fc =i One can not suppose that p > 0, since otherwise the inequality Vh(x) > 0 would be dependent and on the ground of the supposi¬ tion made would not belong to the system (1). By supposing that p < 0, one will admit ph = — p and one will obtain the identity <7 Y PkVk(x) = 0 where p k > 0, ( k = 1, 2, . . . , a) k = 1 which is contrary to the hypothesis. We have supposed that the theorem introduced be true in the case of m — 1 variables. As the equation (8) can not be reduced into an identity, one concludes that the system of inequalities (7) defines a domain Bu in m—1 dimensions. Moreover, by indicating with (« 11 , U21 , • • • , U m - 1 , 1 ) , (u 12 , U 22 , . . . , Ujn — 1,2)5 ■ ■ ■ j (^ltj • • • 5 — l,t) ( 9 ) the points which characterised t edges of the domain B k , one will determine any point ( u ) of this domain by the equalities t Ui = Y PkUik where p k > 0, k =1 (k = 1,2,.. ., t; i = 1,2,.. . ,m - 1) (10) One will make correspond to the points (9) the points ((r)=((ln(2r,...,U), (r = 1,2 , . . . , t ) (11) 28 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series by determining them with the help of equalities (6) and (9). The points obtained (11) characterise t edges of the domain R belonging to the face Ph. Any point ( x ) be¬ longing to the face Ph will be determined, on the grounds of (6) and (10), by the equalities t x i = ^2 PkZik where p k > 0. (k = 1,2, . . ., t; i = 1,2, . . ., m) k = 1 ( 12 ) Let us notice that all the points (11) verify the equa¬ tion Vh(x) = 0 (13) and satisfy the conditions Vk(x) > 0. (k = 1,2, . . . , a) One obtains thus the equalities VhUr)> 0 (r = 1,2, . . . , t; k = 1,2, . . . , a) (14) The face Ph being in m-1 dimensions, the equalities (14) would define the coefficients of the equation (13) to a close by common factor. Let us suppose that one had determined this way all the faces Pl,P 2 ,...,Pa (15) in m-1 dimensions of the domain R. Let us suppose that the points (€fc) = (€ifc,€ 2 fc, €mfc) (fc = 1,2,. . ., s) (16) characterise the various edges of the domain R belonging to the various faces (15). God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 29 Monograph Translations Series G. F. Voronoi (1908) By indicating S x i = ^2 pk^ik where p k > 0, (k = 1, 2, . . . , s; i = 1,2, . . ., m) k = 1 (17) one obtains a set of points which all belong to the do¬ main R. I say that any point ( x ) belonging to the domain R can be determined with the help of equalities (17). One can suppose that the point ( x ) does not belong to any one of the faces (15), since any point belonging to them can be determined with the help of equalities ( 12 ). By supposing that one had the inequalities Vk(x) > 0, (k = 1,2, . . . , a) let us arbitrarily choose a point (£ r ) among those of the series (16) and let us admit x'i = Xi — p^i r where p > 0. (i = 1 , 2 ,..., m) (18) So long as the parameter p is sufficiently small, one will also have Vk(x') > 0 . (k = 1,2,. . ., < t ) By making the parameter increase in a continuous manner, one will determine with the help of equalities (18) a point ( x ') verifying an equation Vh(x') = 0 and satisfying the condition Vk(x') > 0 . (k = 1 , 2 ,. . ., < t ) The point obtained ( x ') belongs to the face Ph, there¬ fore one can declare t x'i = Yj p' k £,ik where p' k > 0. (k = 1,2, . . . , f; i = 1,2, . . . , m) k = 1 30 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series By virtue of (18), it becomes t Xi = p£,ir + ^2 P'kZik where p > 0, p' k > 0. k =1 (k = 1,2, ... ,t-,i = 1,2, ... ,m) It remains to demonstrate that the domain F is in m dimensions. Let us notice that all the points determined by the equalities (17) with condition Pk > 0 (k = 1,2, . . ., s) are interior to the domain R. In effect, all the points (16) verify the inequalities yn^k)> o. (k= 1,2.s; fa = 1,2,...,a) (19) By multiplying these inequalities by pk, let us make the sum of inequalities obtained; one will have, because of (17), Vh(x) = ^2 PkVh(^k) > o. (h = 1,2, . . . , a) k = 1 By virtue of (19), one will have the inequality y h (x) > 0, (h = 1, 2, . . ., s) (20) so long as the numbers 2 /h(£i), ^(£2), • • • , yh(Cs) do not cancel each other out. One can not suppose that the equalities yn{£,k) = 0 (k = 1,2, . . ., s) holds, because otherwise all the equations Vi(x) = 0, 2/2(2:) = 0,. . ., y CT ( x) = 0 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 31 Monograph Translations Series G. F. Voronoi (1908) would be of proportional coefficients, which is contrary to the hypothesis; therefore one will have the inequali¬ ties (20), and it follows that the domain R is of m di¬ mensions. We have demonstrated that the condition (£) is suf¬ ficient for the domain R to be of m dimensions. It is easy to demonstrate that this condition is necessary. We have defined in Number 10 the faces in m — 1 dimensions of the domain R. This definition can be gen¬ eralised. Definition. One will call face in p dimensions of the do¬ main R (p = 1,2, . . ., m — 1) a domain P(p) formed from points belonging to the domain R and verifying a system of equations y k (x) = 0, (k = 1,2,...,r) (21) provided that these equations define a domain in p di¬ mensions composed of points which, all, do not verify any other equation y T + i(x) = 0, . . ., y a (x) = 0. Let us choose among the points (16) all those which verify the equations (21). By indicating with f,k — (f,lk j f,2k j • • • j f,mk) j (k — 1,2, . . . , t) one will declare t Xj = Y] Pkf,ik where p k > 0. (k = 1,2,... ,t\i = 1,2, ... ,m) k = 1 ( 22 ) It is easy to demonstrate that any point ( x ) belong¬ ing to the face P(p) can be determined with the help of equalities (22). 32 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series Corollary. Each face of the domain R is a set of points determined by the equalities (22) provided that any point belonging to it could not be determined by the equalities S x i = ^2 Pk^ik where p k > 0, k = 1 (k = 1,2 ,,s m ,i = 1,2 ,,m) unless all the parameters pt+i , Pt+ 2 , ■ ■ ■ , p s do not cancel each other. Any point belonging to the domain R either is in¬ terior to the domain R or is interior to a face of that domain. Let us suppose that the point ( x ) be interior to a face P(p) of the domain R which is formed from all the points determined by the equalities (22). I argue that one can always determine the point (x) by the equalities (22) provided that Pk > 0. (k = 1,2, . . ., t) To demonstrate this, let us indicate t Pi — ^ ' f,ik- (i — 1 ) 2 ,..., m ) k = 1 The point (a) is interior to the face P(p). By admitting x\ = Xi — poii where; p > 0, (i = 1,2m) (23) one obtains a point (x ' i ) which will be interior to the face P(y) so long as the parameter p will be sufficiently small; it follows that t x'i = P'k&k where p’ k > 0. k = 1 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 200 7 33 Monograph Translations Series G. F. Voronoi (1908) (k = 1,2, . . ., t; i = 1,2, . . ., m) By virtue of (23), one obtains t x i = *52 (p + p'k)€ik, (i = 1, 2 > • • • > m) k =i and by making P + Pk = Pk, (k = 1,2, . . ., t) one will have t Xi = Pkiik where p k > 0. k = 1 (k = 1,2, . . . ,t; i = 1, 2, . . ., m) Let us notice that by making p = m and t = s, one will indicate with the symbol P(m ) the domain /?; one concludes that any point ( x ) which is interior to the do¬ main R can be determined by the equalities S Xi = ^2 PkZik where p k > 0. k = 1 (k = 1,2, ... ,s\i = 1,2, ... ,m) On the correlative domains. Definition. Let us suppose that a domain R be deter¬ mined with the help of inequalities PlkXl + P2kX2 + • • • + PmkXm >0. (fc = 1,2 , . . . , tj) One will call correlative to the domain R the domain 1Z which is formed from all the points ( x ) determined by the equalities <7 Xi = ^2 PkPik where p k > 0. (k = 1,2, . . ., a\ i = 1 , 2, . . ., m) k= 1 ( 1 ) 34 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series I say that the domain 1Z will be in m dimensions, if the domain R does not possess points verifying the equations PlkXl + PlkX 2 + PmkX m = 0, ( fc = 1, 2, . . . , (T) the point x\ = 0, X 2 = 0,. . ., x m = 0 being excluded. In effect, if all the points of the domain 1Z verified the same equation + £ 2X2 + • • • + £m,Xm = 0 , one would have the equalities flPlk + f,2P2k + • • • + f,mPmk =0, ( fe = 1, 2, . . . , £7) by virtue of (1), which is contrary to the hypothesis. Theorem. By supposing that the domain R be formed from all the points (x) determined by the equalities S aq = ^ Pkf,ik where p k > 0, (k = 1, 2, . . . , s; i = 1,2, . . ., m) fc=i ( 2 ) one will define the correlative domain 1Z with the help of inequalities. flkXl f,2kX2 + • • • + f,mkX rn Z 0- (k — 1,2,..., s) (3) Let us indicate by 7 Z 1 the domain determined with the help of inequalities (3). On the ground of the supposition made, all the points (£ll, £21, ■ ■ ■ , £m.l), (€l2, £22, ■■■, £m2 ),■■■, (£l s 5 f, 2 s 7 ■ ■ ■ 7 f,ms ) characterise the edges of the domain R, and one will have the inequalities Plhf,lk d - P2hf,2k Pmhf,mk God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 35 Monograph Translations Series G. F. Voronoi (1908) (h= l,2,...,a;h= 1,2,(4) We have seen in Number 10 that each face Ph in m — 1 dimensions of the domain R is characterised by the points (£l 1) £21 j • • • j 1 ) ) (£l2 j £22 j • • • 1 ( 1112)1 • • • ) (£ It > ^2t) • • • > t) which verify the equation Vh = 0 (5) of the face Ph- One obtains the equalities Plhlilk P P2h£,2k + • • • + Pmh£,mk = 0 (fc=l,2,...,t) which define the coefficients pm,P 2 h, ■ ■ ■ ,Pmh of the equa¬ tion (5) to an immediate common factor. One concludes, by virtue of the definition established in Number 9, that the point (pih,P 2 h, ■ ■ ■ ,Pmh) charac¬ terises an edge of the domain 7Z'. By attributing with the indice h the values 1, 2, . . . , a, one obtains a series (Pll>P21, • • • j P ml) i (Pl2,P22, • • • j Pm 2) j • • • , (Pier , P2cr, • • • , Pmcr) of points which characterise different edges of the do¬ main 1Z'. I argue that the domain 1ZJ does not possess other edges. To demonstrate this, let us suppose that a point Pi, p 2 , • • • , Pm characterises an edge of the domain 1Z'. One will have the equalities Pl^lh + P2^2h + • • • + Pm^mh = 0, (h = 1,2, . . . , t) (6) which define the coefficients Pi,P 2 i ■ ■ ■ iPm to a nearby common factor, and one will have the inequalities PlClh + P2^,2h + • • • + P rri£,mh ^ 0. (fc=l J 2,...,s) (7) 36 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series Let ( x ) be any point of the domain R. One will de¬ termine the point ( x ) with the help of equalities (2). By multiplying the inequalities (7) with p/, and by making the sum of inequalities obtained, one will have, because of (2), PlX± + P 2 X 2 + • • • + Pm^Cm ^ 0 • One concludes that the inequalities - PiX 1 - P 2 X 2 - ■ ■ ■ - PmXm > 0 and PlkXl + P2kX 2 + • • • + PmkXm — 0 ; define a domain which is not in m dimensions. By virtue of the fundamental theorem of Number 10, one will determine in this case positive values or zeros of parameters p,p\,...,p a which reduce the equation ~p(pixi + P2X2 + • • • + Pm^m )+ cr ^2 Pk (PlkXl + P2kX2 + • • • + PmkXm ) = 0 k = 1 into an identity. It follows that Pi = V] — Pik where — > 0.44 ti P P (k = 1,2,. . ., ct; i = 1,2, . . ., m) By substituting (6), one will have ^ ' ( S,±hP±k S,2hP2k £mhPmk) — 0. k = 1 P (h = 1,2,. ..,t) By virtue of (4), one finds Pk (£,lhPlk + £,2hP2k + •••+€ mhPmk ) — 0* God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 37 Monograph Translations Series G. F. Voronoi (1908) (h — 1,2 , }t', k — 1,2,.,.,cr) Let us suppose that > 0, then f,lhPlk f,2hP2k “H • • • “1“ £,mhPmk — 0> (hi — 2, . . . , t) therefore the coefficients Pi, P 2 , ■ ■ ■, Pm, by virtue of (6), are proportional to the coefficients pik, P 2 k, ■ ■ ■ , Pmk] it follows that the points (pik, P 2 k, ■ ■ ■ , Pmk) and (pi,P 2 , ■ ■ ■ , Pm) characterise the same edge of the domain 1Z'. By virtue of the fundamental theorem in Number 10, all the points of the domain 7 Z' will be determined by the equality (1), this results in that the domains 1Z and 7 Z' coincide. Corollary. Let us suppose that a face P(p) in p dimen¬ sions of the domain R be determined by the equations PlkXl + P2kX 2 + • • • + PmkXm = 0, ( k = 1,2, . . . , T) and that any point ( x ) belonging to the face P(p) be de- termined by the equalities t Xi = ^2 Pkf,ik where p k > 0. k = 1 (k = 1, 2 , . . . , t; i = 1, 2 , . . . , m) The correlative domain 1Z will possess a correspond¬ ing face J3(m — p) in m — p dimensions determined by the equations ftlkX 1 f f 2kX 2 “H • • • “H f,mkXm — 0 (^ — 1,2, . . . , t) and any point ( x ) belonging to the face B(m — p) will be determined by the equalities T Xi = ^2 pkPik where p k > 0. k = 1 (k = 1,2, . . ., t; i = 1,2, . . ., m) 38 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series Definition of domains of quadratic forms correspond¬ ing to the various perfect forms Let us consider any one perfect quadratic form cp. Let us suppose that all the representations of the minimum of the form <p make up the series (hi, hi, ■ ■ ■ , Ini') , (l 12, 122, ■ ■ ■ ,ln2), ■ ■ ■ , (hsjhs, ■ ■ ■ ,lns )• (1) By indicating A k — 11 k X 1 “1“ 12 k X 2 • • • “1“ l nkX n , (k — ( 2 ) one corresponds to the series (1) a series of linear forms Ai, A 2 , . . ., A s . Let us consider a domain R of quadratic forms de¬ termined by the equality S f(xi,x 2 ,. . . ,x n ) = Pk^l k=l with condition that Pk > 0. (k = 1,2, . . ., s) One will say that the domain R correspond to the perfect form <p. Let us notice that the domain R is in n ( n + 1 '> dimen¬ sions. By supposing the contrary let us suppose that all the quadratic forms belonging to the domain R verifies a linear equation ^(/) = Y Pi 3 ai 3 = °' God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 200 7 39 Monograph Translations Series G. F. Voronoi (1908) On the ground of the established definition, one will have the equalities ^(A|) = 0 (k= 1,2,....a) or, that which comes to the same thing, because of (2), Pijlikljk — d ( k — 1 j 2, . . . , S ) which is impossible, the form ip being perfect. On the ground of what has been said in Number 9— 14, the domain R possesses s edges characterised by the quadratic forms A?, A3,..., Aj. (3) Let us suppose that one had determined all the faces Pi, F > 2, ■ ■ ■ , P a in n< , n + 1 '> — i dimensions of the domain R. Each face Pk can be determined by two methods: 1. All the quadratic forms belonging to the face Pk verify an equation ^k(f) = = 0 which can be determined in such a way that the inequal¬ ity **(/)> 0 held so long as the form / belonging to the domain R is exterior to the face Pk- 2. By choosing among the quadratic forms (3) these \2 \2 \2 Ai, A 2 , • • • , A t which verify the equation (4), one will determine all the quadratic forms belonging to the face Pk by the equali¬ ties t f(x !,X 2 , ■ ■ ■ ,X n ) = ^PkXl, k = 1 40 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) where Monograph Translations Series Pk> 0. (fc = 1,2,....t) By virtue of the theorem of Number 14, the domain R can be considered as a set of points verifying the in¬ equalities **(/)> 0. (k=l,2,...,a) On the extreme quadratic forms Let us indicate by M(aij ) the minimum and by D(aij) the determinant of a positive quadratic form Jj aijXiXj ■ The positive quadratic form ^/.D (aij ) determinant 1 and will possess the minimum J2 a ij x i x j will be of M (a jj ) '{/ D (a t -j ) M(aij). Let us examine the various value of the function M(aij) which is well determined in the set (/) of all the positive quadratic forms in n variables. Definition. One will call extreme J a positive quadratic form aijXiXj which enjoys the property that the corre¬ sponding value of the function JA(aij ) is minimum. Let us notice that the function M(aij) does not ch¬ ange its value when one replaces quadratic form Jj aijXiXj by a form of proportional coefficients. By attributing to the coefficients of the extreme form Jj aijXiXj variations e ij = e ji (* = 1,2,. . ., n ; j = 1,2, . . ., n) | See the Memoire of Mr. ’s Korkine and Zolotareff, Sur les formes quadratiques [On the quadratic forms], Math- ematische Annalen V. VI, p. 368 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 jl Monograph Translations Series G. F. Voronoi (1908) satisfying the condition I j I <- e , (* — 1,2, . . ., n , j — 1,2,. . . , n) (1) e being an arbitrary positive parameter, let us examine the corresponding value of the function M(aij). On the ground of the definition established, one can determine the parameter e such that the inequality M(a,ij + tij) < M(aij) ( 2 ) held with condition (1) and so long as the coefficients eij are not proportional to the coefficients aij. (i = 1,2, . . . ,ra; j = 1,2, . . ., n) Theorem. For a quadratic form QijXiXj to be extreme, it is necessary and sufficient that it be perfect and that its adjoint form be interior to the domain corresponding to the form JfaijXiXj. Let us indicate by Oil, hi, ■ ■ ■ , In l), 0l2, 1 22, • • • 0n2), • • • , (l Is, hs, ■ ■ ■ , l ns) (3) the various representations of the minimum M(aij) of the form JfaijXiXj. Let us consider a quadratic form + peij)xiXj , the parameter p being arbitrary. One can determine an interval — 5 < p < 6 where 0 < <5 < 1 (4) such that all the representations of the minimum of the form J2(aij + peij)xiXj are found among the systems (3) so long as the variations tij satisfy the condition (1). By indicating with AT — f ' (a l y “I - peij)likljk and AT — f a j j l j r. I j k (3) f2 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series the minima of forms Yl( a ij + p€ij)xiXj and J2 a ij) x i x j and with D' and D their determinants, one will have + peij) ^D 7 A4 (o-ij ) a ij likljk sfD By virtue of (2), one obtains the inequality “I - P^-ij)likljk ^ Edijlikljk r <fly </T> or, that which comes to the same thing, v- / D' P 2_ _J e ij likljk < FT W — 1 D ( 6 ) This declared, let us suppose that the form J2 a ij x i x j be not perfect. One will determine in this case the variations eij such that the equalities X ^ ^ijljkl jk — Oj held. By virtue of (6), one will obtain the inequality D' > D. By developing the determinant D' into a series, one will have the inequality dD The parameter p being arbitrary satisfying the con¬ dition (4), it is necessary tht E dD ~ ij da = 0. God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 f3 Monograph Translations Series G. F. Voronoi (1908) Mr.’s Korkine and Zolotareff have demonstrated J that in this case one will always have the inequality E d 2 D daij dcikh < 0, therefore the inequality (7) is impossible. We have demonstrated that the form c p = Y) a ij x i x j has to be perfect. Let us suppose that the domain R corresponding to the perfect form c p be determined by a inequalities > °. (r = 1,2, . . . , a) On the grounds of these inequalities, one will have ^r(A^) = ^P^likljk >0. (k = 1,2, . . ., s; r = 1, 2, . . ., a) ( 8 ) Let us declare eij = tp\ r ^ where t>0. (i = 1, 2 , . . ., n; j = 1, 2 , . . ., n) By virtue of (6), one will have P* ^2 p i r j )likl l k < M ^ ^ • ( 9 ) Let us attribute to the parameter p a positive value satisfying the condition (4), by virtue of (8) and (9) there will arrive D 1 > D. By developing the determinant D' into a series, one obtains the inequality dD ( r ) ^ da + (pty E (r) (r) PijPih- d 2 D daij dakfi + ... > 0 . | Mathematische Annalen, V. XI, p. 250 44 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series The positive parameter p being as small as one wish, it follows that I>S’ dD dan > 0 . (r = 1,2, . . . , a) It is thus demonstrated that the form J2 ~§^~ x i x j > ad- joint to the form ip, is interior to the domain R. I argue that in this case the perfect form p> will be extreme. By supposing the contrary, let us suppose that the inequality -)- eij) > Af(ajj) (10) be verified by any one system of variations (i = 1,2, . . . , n; j = 1, 2 , . . . , n) satisfying the condition (1) however small the parameter e may be. By virtue of (10), one obtains ( n rw \ ^tijlikljk > M I y—- 1 ; (fe = 1,2, . . ., s) (11) the inequality obtained has to hold whatever may te the value of the index k = 1, 2, . . ., s. By indicating (z — 1, 2, . . . , n ; j — 1,2, . . . , n) (12) let us examine the quadratic form ip 0 (x 1 , x 2 , ■ ■ ■ , X n ) =^2(aij + T] ij )xiX j . (13) By virtue of (12) the form <p 0 is of determinant D. By choosing the parameter e sufficiently small, one can suppose that \Vij \ < r), (i = 1, 2, . . . , n; j = 1,2, . . . , n) (14) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 45 Monograph Translations Series G. F. Voronoi (1908) r) being a positive parameter as small as one would like. By virtue of (5), (11) and (12), one obtains y^Vijlikljk > 0- (fc = 1,2, . . ., s) (15) By developing the determinant D of the form (13) into series, one will find D + l >« 5 ^- + h » = d ' (16) In this equality the remainder R 2 verifies an inequal¬ ity |i?2| < tfP, P being a positive number not depending on the param¬ eter r] so long as rj < 1. By virtue of (16), one obtains < V 2 P- (17) E 7 ^ dD dai We have suppose that the quadratic form -§P-XiXj, adjoint to the form ip, be interior to the domain R. On the ground of that which has been said in Number 13, one will determine the form J2 x i x j with the help of the equality E dD - - XiXj u a ij E ? kX * k — 1 (18) where Pk > 0. (k = 1, 2, . . . , s) (19) The equality (18) can be replaced by the following ones: 0D_ d a ij S ^ ^ Pk likljk- (^ 1 ^ , . . . , 71 , j — 1,2, . . • , Tl ) k — 1 46 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series By multiplying these equations by 77 ^ and by adding up the equalities obtained, one will have dD O&ij ^ ' Pk ^ ' Pi j likl jk • k = 1 ( 20 ) By virtue of (15), (17) and (19), one obtains the in¬ equalities _ p 0 ) ( Pijlikljk ^ P 5 (A) — l,2,...,s) Pk therefore one can admit ) ' Pijlikljk — T~kP j (k — 1 j 2,_..., S ) (21) and the positive numbers or zeros Tk (k = 1,2 , . . . , s) will not exceed fixed limits which do not depend on the pa¬ rameter 77 . After the definition of perfect forms, the equations (21) admit only a single system of solutions. By effecting this solution of equations ( 21 ), one obtains Pij = Tijp 2 (i = 1, 2, . . ., n; j = 1,2, . . ., n) where I T ij I < T, ( 7 = 1, 2, . . . , ra; j = 1,2,..., 71 ) T being a positive number which does not depend on the parameter 77 ; therefore one will have the inequalities \pij\<p 2 T. (i = 1,2, . . . , ra; j = 1, 2, . . . , ra) (22) This stated, let us take any one positive fraction 1 ? and declare By virtue of (14), one will have \P I "S' rj-i 5 ( 7 = 1,2,... ,n; j = 1,2,...,ti) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 f7 Monograph Translations Series G. F. Voronoi (1908) and because of (22), it will become 0 ' \Vij\ < -rjT- (* = 1,2, . . ., n; j = 1,2, . . ., n) By admitting V 2 tiij < p 1 one will have, because of (22), i ? 4 \Vij\ < ~p, (i = 1,2, ■ ■ ■, n; j = 1, 2, . . . , n) and so on. One will obtain in this manner the inequalities 0 2k \Vij \ < ~p~ ( l= 1 ,2, . . ., n; j = 1, 2, . . . , ra; fc = 0, 1,2, . . .) it follows that Vij = 0. (i = 1,2, . . ., n; j = 1,2, . . ., n) By virtue of (12), one obtains therefore the coefficients are proportional to the co¬ efficients ciij (i = 1,2, . . . , n; j = 1, 2, . . . , n), which is con¬ trary to the hypothesis. Properties of the set of domains corresponding to the various perfect forms in n variables. Any perfect form cp will be transformed by an equiv¬ alent perfect form p 1 with the help of any substitution S of integer coefficients and of determinant ±1. Let us indicate by R and R' the domains correspond¬ ing to the perfect forms <p and ip' and by T the substitu¬ tion adjoint to the substitution S. 48 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series One will easily demonstrate that the domain R will be transformed into an equivalent domain R' with the help of the substitution T. One concludes that there exists a finite number of domains equivalent to the domain R. Let us indicate by (f?) the set of all the domains cor¬ responding to the various perfect forms in n variables. The set (f?) can be divided into classes of equivalent domains. On the ground of that which has been previously said, the number of different classes of the set (f?) is equal to the number of classes of perfect forms in n vari¬ ables. Theorem. Let us suppose that a quadratic form f be interior to a face Pin) in p dimensions of the domain R(p= The form f will belong only to the domains of the set ( R ) which are contiguous through the face P(p). Let us suppose that the domain R be characterised by the quadratic form A?, A|.(1) and that the face P(p) in p dimensions of the domain R be characterised by the quadratic forms (2) In the case p = n dH-i) , one w jh admit t = s, and the symbol P ^ n ( n + 1 '> ^ will indicate the domain R. The quadratic form / being interior to the face P(p ), God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 Monograph Translations Series G. F. Voronoi (1908) one will have the equality t f(xi,x 2 , . . ., X n ) = Pk^l where p k > 0. (k = k = 1 1,2 (3) Let us suppose that the same form / belonged to another domain R' of the set ( R ). Let us suppose that the domain R' be characterised by the quadratic forms \' 2 \' 2 v 2 (4) and that the form / be interior to the face P'(u') of the domain R’ characterised by the quadratic forms \' 2 V 2 A i , a 2 , A' (5) One will have, on the ground of the supposition made, f(xi,x 2 , • • • , x n ) = ^ Ph x 'h 2 where p' h > 0. (h = 1,2, . . . , r) h= 1 ( 6 ) This declared, let us indicate by <p and ip 1 the perfect forms corresponding to the domains R and R' and sup¬ pose, for more simplicity, that the minimum of forms <p and </?' be M. By indicating with the symbol (/,/') the result (/, /') = Y2 ai i a 'ij > from two quadratic forms f (.X 1 , 3? 2 , • • ■ , X n ) — ^ ' Q'ij X iX j and f(x i, x 2 , . . . , x n ) = ^2 aljXiXj , let us examine two results and (/,<£>'). 50 February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series By virtue of (13), one obtains t t (/, V) = Y Pk{T>, and (/, if') = Y Pk(<p', A£ fc=1 /c=l By virtue of (6), one obtains (/>¥>) = Y PhiV’Xh 2 ) and (/>¥>') = Yp'hW' X 'h h=l h=l Let us notice that ( <p,\l) = M and ( cp', X 2 k ) > M ; (fc = 1,2,..., (v,\' h 2 )>M and (^',A' h 2 ) = M. (fc = 1,2, . . ., ct From equalities (7), one derives (/, ¥>') - (/, V) = Y P k [(^'> A fc) _ (V. A fc)l > k = l and by virtue of (3) and (9) there comes (/, ¥>') - (/, ¥>) > 0. From equalities (8), one derives (/. ¥>') - (/. ¥>) = Y Ph [(¥>'. A 'h 2 ) - (V. A 'h 2 )] . h = l and by virtue of (6) and (10), one will have (/, ¥>') - (/, ¥>) < 0. It follows that (/, ¥>') = (/, V 3 ), God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, ( 7 ) )• ( 8 ) ) ( 9 ) ( 10 ) ( 11 ) ( 12 ) 2007 51 Monograph Translations Series G. F. Voronoi (1908) and the equalities (11) and (12) give (¥>', Afc) = (<p, X 2 k ), (k = 1,2, . . .,t) = (V,K 2 )- (h= 1,2,... j t) By virtue of (9) and (10), there arrive (<p,\' h 2 ) = M, (h = l,2 J ... J r) (V',A l) = M. (k = 1,2,...,*) (13) (14) By virtue of equalities (13), the quadratic forms (5) are found among those of the series (1). By virtue of (14), the quadratic forms (2) are found among those of the series (4). I argue that in this case the series (2) and (5) contain the same forms. To demonstrate this, let us suppose that all the forms belonging to the face P(/x) verify the equations *l(/) = 0, * 2 (/) = 0,-.., * r (f) = 0 and that any form belonging to the domain R verifies the inequalities *i(/)> 0, * 2 (/)>0,..., *r(/)>0. (15) By virtue of (6), one will have p'^iiX 1 ! 2 ) + p' 2 *i(\' 2 2 ) + ■ ■ ■ + p't(K 2 ) = 0, (*= l,2,...,r) and because of (15), one finds 'MAk 2 ) = 0; (* == 1,2,.. .,r;h = 1,2,. . . , r) therefore all the forms of the series (5) belong to the series (2). In the same way, one will demonstrate that all the forms of the series (2) belong to the series (5). One concludes that the faces B(p.) and P'(v) coincide, therefore the domains R and R' are contiguous through the face P(p,). 52 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series Corollary. A quadratic form which is interior to a do¬ main of the set ( R ) can not belong to any other domain of that set. 21 Theorem. Let us suppose that to a face P(y) of the do¬ main R belong positive quadratic forms. In this case, the number of domains of the set ( R ) contiguous through the face P(j- 1 ) is finite. Let us indicate by R, Ri, i?2, • • • the domains of the set (f?) contiguous through the face P(/x). Let ¥b <Pl,<P2, ■ ■ ■ be the corresponding perfect forms having the minimum M. On the ground of the supposition made, one positive quadratic form / will be interior to the face P(p). We have demonstrated in the previous number that (/,¥>) = (/,¥>i) = (/, ¥> 2 ) = •■ ■ (16) It is easy to demonstrate that the number of perfect forms having the minimum M and verifying the equali¬ ties (16) is finite. Algorithm for the search for the domain of the set (R ) contiguous to another domain by a face in n ^ n 2 +1 ' ) — 1 di¬ mensions Let (p (x 1, X2 , . . . , X 77 , ) — Oij X}X God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 53 Monograph Translations Series G. F. Voronoi (1908) be a perfect form having the minimum M the various representation of which make up a series (^11, l2h ■ • • 0l2, 1 22, • • • , 1 712 ), ■ ■ ■ , (l Is, 12s, ■ ■ ■ , Ins)- (1) Let us suppose that a face P of the domain R cor¬ responding to the perfect form ip be determined by the equation *(/) = J2Pij a H = 0 and by the condition *(/) > o which is verified by any quadratic form belonging to the domain R. Let us suppose that the face P be characterised by the quadratic forms \2 \ 2 w Ai, A 2 , • • • , A t ( 2 ) where k — 1 1 k -E 1 12k -E2 “ 1 “ • • • “ 1 “ lnk-E n • ( h — 1 , 2 ,. . . , S ) On the ground of the supposition made, one will have the equalities ^ Pijhkljk = 0 (k = 1,2, . . . , t) (3) which define the coefficients Pij (i = l,2,...,n;j = 1,2, . . . , n) to an immediate common factor. Let us suppose that the face P could belong to the other domains of the set (R). Let us indicate by R' a similar domain. Let ip' be the perfect form corresponding to the domain R'. By virtue of the supposition made, the quadratic form (2) belong to the domains R and R'. It results in that the systems (In, 121, ■ ■ ■ , Ini), (l 12, 122, [• • • , ] ln2 ) , ■ ■ ■ , (l It, 12t, ■ ■ ■ , Int) (4) 54 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series corresponding to the forms (2) represent the minimum of forms ip and ip 1 . Let us suppose, for more simplicity, that the forms ip and ip' had the minimum M. One will have the equalities ^ ' ciijlikljk —— Mi and ^ ' &ijlikljk —— Ml , (k — 1, 2,. . ., t) (5) by putting <^'(^ 1 ,^ 2 , • • • , x n ) = ^aGxiXj. From equation (5), one gets ( a ij — a ij)likl jk = o, (k = 1,2, . . . , t ) and by virtue of (3), it becomes a 'ij = a ij + PPij- (i = 1,2, . . ., n; j = 1,2, . . ., n) (6) Let us indicate <F(x 1 ,x 2 , ■ ■ ■ ,x n ) = Y^Pij x i x j- By virtue of (16), one obtains (p'(x 1 ,x 2 , ■ ■ ■, x n ) = cp(x 1 ,x 2 , ■ ■ ■ ,x n ) + p'&(x 1 ,x 2 , ■ ■ ■ ,x n ). ( 7 ) This stated, let us choose in the series (1) a system (Zi h, l 2 h, • • •, l n h ) which does not belong to the series (4). As ’F ( Z lh j • • • j lnh) — Af </? {llhi l 2 ft, j • • • j lnh ) ^ Af and (l lh i 12h i ••• i lnh) ^ 0, one deduces from the equality SP (llh : l2hi ■ ■ ■ : l nh) = SP (Z lh > 12h :•••■> l nh) pAt (l lh , 12h > • ■ ■ > Z nft.) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 55 Monograph Translations Series G. F. Voronoi (1908) the inequality p > 0. The supposition p = 0 being obviously impossible, one obtains P> 0 , and it follows that i P (l lh i l2h i • • • i Inh ) ^ Ml . Let us indicate by (I'l , *2, ■ ■ ■ , O. (M. ■ ■ ■ . C), ■ - ■ , (8) all the representations of the minimum of the perfect form ip' which are not found in the series (4). By virtue of (7), one will have cp'(l[ k \l {k \...,l^) = cp(l[ k) ,l {k \ ■■■,l (k) ) + p^(l {k \l {k \. . . , ( k= 1,2,....r) which results in <p(l[ k) ,li k) ,.. .,lW) > M and *(l[ k) ,l ( 2 k) ,...,lW) <0. (k= l,2,...,r) (9) The value of the parameter p will have for expression _ cp(l{ k \l {k \...,l { n k) )-l (*)■ (k 1 , 2 , ■ Let us examine any one value of the function v(x 1,X 2 , • • •, X n ) - M A'(x 1 ,X 2 , • • • , X n ) ( 10 ) determined with the condition '&(x 1 ,x 2 ,. . . , x n ) < 0 . ( 11 ) 56 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series I argue that one will have the inequality <p(xi,x 2 , ■ ■ ■ , x n ) -M -/-\- P ' W(Xi,X2, ■ ■ ■ ,X n ) ~ Let us suppose the contrary. By supposing that <p(x 1,X 2 , • • • , X n ) -M -iT7-\ < P, V(xi, X2, ■ ■ ■ , Xn) one will find, because of (11), <p(xi,x 2 , ■ ■ ■ ,Xn) + P'&(xi,X 2 , ■ ■ ■ ,Xn) < M or, that which comes to the same thing because of (7), <p'(x 1,X2, • • • , Xn) < M, which is contrary to the hypothesis. We have arrived at the following important result: There exists only a single domain R' contiguous to the domain R through the face P. The corresponding perfect form p 1 will be determined by the equality (7) provided that the parameter p presents the smallest positive value of the function (10). Let us notice that by virtue of (3) and (9), all the quadratic forms belonging to the domain R’ verify the inequality *(/)< 0 . One concludes that the domains R and R' are found from two opposite sides of the plane in n bH~ 1 ) _ ^ dimen¬ sions determined by the equation *(/) = o. The smallest positive value of the function (10) can be obtained with the help of operations the number of which is finite. God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 57 Monograph Translations Series G. F. Voronoi (1908) The whole problem is reduced to the preliminary stu¬ dy of a system (Zi, Z 2 , . . . , l n ) of integers verifying the in¬ equality 'i'ili,h, ■ ■ ■ ,l n ) < 0 and satisfying the condition that the quadratic form <Po(xi,x 2 , ■ ■ ■ ,x n ) = <p(x 1} x 2 , ■ ■ ■ ,x n ) + p 0 '&(x 1 ,x 2 , ■ ■ ■ ,x n ), where one has admitted (P (l 1 1 12 i • ■ • t In') — M be positive. One will determine in this case all the systems (aq, x 2 , ■ ■ ■ , x n ) of integers verifying the inequality <p(x 1 ,x 2 , ■ ■ ■ ,x n ) > M (12) the number of which is finite, and one will find among these systems all those which define the smallest value of the function (10). Let us indicate by R, as we have done in Number 2, the upper limit of values of the parameter p. The problem is reduced to the study of a system (Z!, l 2 , . . . , l n ) °f integers verifying the inequality ‘P’ill, I 2 , ■ ■ ■ , In) + R'P (ll > I 2 , ■ ■ ■ j In) <. M. (13) It can turn out that the equation <p(x i,x 2 , ■ ■ ■ ,x n ) + R^>(x 1 ,x 2 , ■ ■ ■, x n ) 0 will be verified by integers, one will determine them with the help of equations 3 V + dx dxi 0. (i = 1, 2, . . . , n) In the case where these equations can not be verified by any one system of integers, one will study the values of linear forms Op _l_ R d £ dxi dxi (i = 1,2, ... ,n) 58 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series and one will determine as many as one wish of the sys¬ tems of integers verifying the inequality (13). By supposing that a system of integers (Zi, l 2 , ■ ■ ■ , l n ) verifying the inequality (13) were determined, one can look for the smallest positive value p of the function (10) with the help of the following procedure. The inequality (12) can be put under the form cp(xi,x 2 , ■ ■ ■ ,x n ) (l - ^ [<p(x 1 ,x 2 , ■ ■ ■ ,x n ) + RA'(xi, X 2 , ■ • • , Xn)] < M, and as <p(xi,x 2 , ■ ■ ■ ,x n ) + R'F(x 1 ,x 2 ,. . ., x n ) > 0, it becomes tp(xi, X 2 , ■ ■ ■ , Xn) (1 - < M, or differently <p(xi, X 2 , ■ ■ ■ , Xn) < M R R — Po Among the systems of integers verifying this inequal¬ ity one will find all the systems (8) searched for. Algorithm for the search for the domain of the set (R ) to which belongs an arbitrary positive quadratic form. Theorem. Any positive quadratic form belongs to at least one domain of the set (R). Let f (x 1 j x 2j ... j x n ) — ) ( a%jxiXj be any one positive quadratic form. Let us choose any one domain R from the set (f?). God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 59 Monograph Translations Series G. F. Voronoi (1908) Let us suppose that the form /, did not belong to the domain (f?). In that case all the linear inequalities which defined the domain R will not be verified. Let us suppose that one had the inequality ^(/) = < °’ (!) Let us indicate by f?i the domain contiguous to the domain R through the face in n ( n + 1 '> _ i dimensions de¬ termined by the equation *(/) = o. By indicating with <p and (pi the contiguous perfect forms corresponding to the domains R and R±, one will have, as we have seen in Number 22, (pi(x 1 ,X 2 , ■ ■ ■ ,X n ) = (p(x!,X 2 , ■ ■ ■ ,X n ) + p^'(x 1 ,X 2 , ■ ■ ■ ,X n ) (2) where p > 0 and SI/(;ei, x 2 , • • • , x n ) = Y)Pij x i x j- Let us examine two results and (f,<p i). By virtue of (2), one will have (/, ¥>i) = (/, ¥>) + P(/> ^), and as, because of (1), (/, = 'YjPijO'ij < °> it becomes (/> <P) > (/, <Pi)- Let us suppose that by proceeding in this manner one obtains a series of domains R, Ri, J?2j • • • • (3) By indicating with ¥>, ¥>i, ¥>2, ■ ■ ■ (4) 60 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series the series of corresponding perfect forms, one will have the inequalities (/,¥>) > (f,T> 1 ) > (f,<P 2 ) > ••• so long as the form / did not belong to the domains (3). By noticing that all the perfect forms (4) possess the same minimum M , one will easily demonstrate that the number of perfect forms (4) verifying the integrality (/,¥>) < P is bounded, whatever may be the value of the positive parameter P. One concludes that the series of domains (3) will necessarily be terminated by a domain R m to which be¬ longed the form / considered. Study of a complete system of domains representing the various classes of the set ( R ). Let R be any one domain of the set (f?). Let us sup¬ pose that one had determined all the domains R, R±, f?2, • • •, R<t (1) contiguous to the domain R through the various faces in n ( n + 1 '> — 1 dimensions, then let us suppose that one had determined all the domains contiguous to the domains (1) and so on. I say that by proceeding in this manner one will come across any domain of the set ( R ) arbitrarily chosen. For example, if one wish to arrive at a domain , one will choose a positive quadratic form / which is inte¬ rior to the domain R and one will proceed as we have done in Number 24. One will determine this way a series of domains R,R',R",...,R M ,R W God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 61 Monograph Translations Series G. F. Voronoi (1908) which are successively contiguous through faces in n ( n + 1 l — 1 dimensions. We have seen in Number 19 that the set ( R ) can be divided into classes of equivalent domains the number of which is finite. Let us find a system of domains representing the var¬ ious classes of the set ( R ). By starting from the domain R, we have determined all the domains R' 1) R 2 ; ■ ■ ■ j Rcr contiguous to the domain R. By not considering equiva¬ lent domains as being different, let us choose among the domain (1) those which are not one to one equivalent and are not equivalent to the domain R. Let us suppose that one had obtained the series R, Ri, R 2 , ■ ■ ■ , Rp~i (2) of domains which are not one to one equivalent. One will study in the same way the domains contigu¬ ous to the domains R, R±, R 2 , • • •, R^-i and one will extend the series (2) by adding to it new domains R/X 5 + ’ ’ ’ 5 Rv 1 which are not one to one equivalent and are not equiva¬ lent to the domains (2). By proceeding in this way, one will always obtain a series R, R±, f?2, • • • j R t 1 (3) which enjoys the following property: the domain belong¬ ing to the series (3) are not one to one equivalent and all the domains contiguous to the domains (3) are equiv¬ alent to them. The series (3) obtained presents a complete system of representations of various classes of the set (f?). 62 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series The study of the series (3) can be facilitated partic¬ ularly by the help of substitutions which transform into itself the domains of the set (f?). Let us suppose that the domain R corresponding to a perfect form p be determined by the inequalities (fc = 1 , 2 , . . ., ct) By declaring ^k(x !,x 2 , ■ ■ ■ ,x n ) = XiXj, (k = 1,2,... ,a) one will determine, as we have seen in Number 22, by the equalities Pk = P + Pk^k (fc = 1,2,..., <x) (4) a perfect forms px, P 2 , ■ ■ ■, Pa- One will call them con¬ tiguous to the perfect form p. Let us indicate by g the group of substitutions which do not change the perfect form p. The perfect forms p\, P 2 , ■ ■ •, Pa being well determin¬ ed by the perfect form p, one concludes that all the sub¬ stitutions of the group g will only permute the forms Pi, P 2, ■ ■ ■ ,Pa- By not considering as different the forms in propor¬ tional coefficients, one can say, by virtue of (4), that the group g will only permute the quadratic forms ^i, A/ 2 , • • • , (5) Let us suppose that one had chosen in this series the forms 'Ll, 'La, . . . , i (6) which enjoyed the following properties: each form of the series (5) will be transformed into a form of the series (6) with the help of a substitution belonging to the group God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 63 Monograph Translations Series G. F. Voronoi (1908) g, the forms (6) can not be transformed one to one with the help of substitutions of the group g. The perfect forms T>k = T> + Pk'&k (k = 1,2, . . . ,p-l) can replace the perfect forms (4), therefore one will de¬ termine only the values of parameters pi, p 2 , ■ ■ ■, p^-i- The corresponding domains Ri, R 2 , ■ ■ ■, R^-i can replace the domains (1). It can come to pass that among the domains R, R±, R 2 , . . . , R^ i are found equivalent domains, one will recognise this with the help of particular methods. On a reduction method of positive quadratic forms. Definition. One will call reduced any positive quadratic form belonging to any one domain R, Ri, R 2 , ■ ■ ■, R T l (1) of a complete system of representations of various classes of the set (f?). Let us suppose that one had determined all the sub¬ stitutions Si,S 2 ,...,S m (2) which transform the domains contiguous with the do¬ mains (1) through the faces in n O+ 1 ) _ ^ dimensions into these domains here. Let / be any one positive quadratic form which is not reduced. One will determine with the help of the algorithm shown in Number 24 a series of domains R,R',R",.. . ,f? (h) 64 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series successively contiguous. Let us suppose that the domain R (h) be the first one which does not belong to the series (!)• With the help of a substitution S' which is found among those of the series (2), one will transform the domain R^ into a domain R k belonging to the series (!)• By transforming the form / with the help of the sub¬ stitution S' into an equivalent form /', one will determine with the help of the form /' a new series of domains Rk > R'k ! ■ ■ • j R (t) and so on. One will determine in this way a series of substitu¬ tions S',S",...,S^ which, all, are found among those of the series (2) and the product s = S'S" ■ ■ ■ S^ of which presents a substitution S with the help of which the form / will be transformed into a reduced form. Let us suppose now that two reduced forms / and /' be equivalent. If one of these forms, for example /, is interior to the domain R k , the form /' will also be interior to it. One concludes that the form / can be transformed into a form /' only with the help of a substitution which transforms the domain R k into itself. Let us suppose that the reduced equivalent forms / and /' be interior to the faces in p, dimensions of domains (!)• In this case one will declare supplementary condi¬ tions for the reduced forms / and /'. After having de¬ termined all the faces in p, dimensions of domains (1), God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 65 Monograph Translations Series G. F. Voronoi (1908) one will choose a complete system of representatives of these various classes. Let us suppose that this system be formed by the faces in p dimensions Pi(ti), -P2<»> • • • > (3) Any positive quadratic form which is interior to a face in p dimensions of a domain of the set (J?) will be equivalent to a form which is interior to the faces (3), one will call it reduced. Two reduced positive quadratic forms which are in¬ terior to the faces (3) will be equivalent only provided that they be interior to the same face and that the sub¬ stitution which transforms one of them into another one also transforms this face into itself. We have arrived at the following result: A reduced quadratic form can be transformed into an¬ other reduced form or into itself only with the help of a substitution which transforms into itself a domain or a face of domains belonging to the series (1). Second Part Some applications of the general theory to the study of perfect quadratic forms On the principal perfect form We will not consider as different the quadratic forms of proportional coefficients, therefore one can arbitrarily choose the minimum value of a positive quadratic form. In that which follows, one will study only the perfect quadratic forms whose minimum is 1. One will indicate by D the determinant of these forms. 66 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series Among the various perfect forms, one form = x\ + x\ + • • • + X 2 n + X1X2 + X1X3 + . . . + X n -lX n t where 1 2 ’ da — 1 , ( i — 1 ; 2 , . . . , Ti ), (i = 1,2 ,n;j = 1,2, ... ,n\i j) and D n ■ 1 2 n One will call principal the perfect form c p. The perfect form cp possesses n ( n + 1 l representations of the minimum 1, which define n ( n + 1 l linear forms Ai = £Ci, A2 = X2, ■ ■ ■ , \n — xji, A n +i — x 1 X21 A n _|_2 — «r 1 X3, . . ., A n(n+l) - X n —1 X n . 2 The domain R corresponding to the perfect form ip is made up of all the quadratic forms determined by the equality "(" + !) 2 ^dijXiXj = ^2 Pk X l where p k > 0. k = 1 (k= 1,2,.. n(n + 2 ~ 1 ) ) From this equality one obtains Pk = aik + a 2k + • • • + a nk so long as k = 1, 2, . . ., n, p k = —dij so long as k > n; The form ip has been given for the first time by Zolot- areff in a Memoire titled: On an indeterminate equation of the third degree (in Russian) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 67 Monograph Translations Series G. F. Voronoi (1908) (i = 1, 2, . . ., n; j = 1, 2, . . ., n; i ^ j) therefore the domain R will be determined by the fol¬ lowing inequalities: dik + & 2 k + • • • + o-nk > 0, (k — 1 , 2, . . ., n) — a ij > 0. (* = 1,2, . . ., n ; j = 1,2,..., n \ i j) By virtue of (1), the perfect form ip possesses n ^ n + 1 l contiguous perfect forms which are determined by the equalities ' Vfc = + PkX k (x 1 ,x 2 , ■ ■ ■ ,x n ), (k = 1,2, ... ,n) < ip k = ip - pkXiXj, ^fc = n + l,n + 2 ,..., n ( n + , (2) (* = 1,2 , . . . ,n-,j = 1,2, . . . , ra; i £ j) Let us find equivalent forms among the perfect forms contiguous to the perfect form ip. To this effect, let us determine the group g of sub¬ stitutions which do not change the form ip. Let us examine, in the first place, the form adjoint to the form ip. One will easily demonstrate that the coefficients of the form adjoint to ip are proportional to those of the form u; = + A 2 + • • • + A ^n+i) • (3) One concludes, by virtue of the theorem of Number 17, that the principal perfect form ip is extreme. The quadratic form uj will have for expression uj = nx\ + nxl + • • • + n x^ — 2x±X2 — 2x1X3 — ... — 2x n -ix n where da — n, (i — 1 , 2, . . ., n j oq j — 1 - 68 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series (i = 1 , 2, . . . ,ra; j = 1 , 2, . . ., n; i ^ j) Let us find all the representations of the minimum of the form u>. The linear forms xi, x 2 , ■ ■ ■ , x n , x-i + x 2 + • • • + x n (4) characterise n + 1 representations of the value n of the form u>. I say that the form u has the minimum n and all the representations of the minimum of the form u> are characterised by the linear form (4). To demonstrate this, let us examine any one value cv(xi, X 2 , ■ ■ ■ , x n ) of the form u. By supposing that none of the numbers x±, X 2 , ■ ■ ■ , x n becomes zero, one will have by virtue of (13) u)(x 1 : x 2 , • • • , x n ) > n, the system x± = 1, X 2 = 1, ... ,x n = 1 being excluded. Let us suppose that any one of the numbers x±, X 2 , • • • , x n does not cancel out and that ■Efc+i = 0) *Efc +2 = 0, . . ., x n = 0; one obtains, by virtue of (3), oo(x i, x 2 , . . ., x k , 0, . . . , 0) = (n - k + l)(xf + x% + . . . + xl) + ^2(x k - x h ) 2 , and it follows that u>(xi,x 2 ,. . . ,x n ) > k(n - k + 1), therefore oo(xi,X 2 ,...,x n )>n so long as k > 2. God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 69 Monograph Translations Series G. F. Voronoi (1908) This stated, let us indicate by G the group of sub¬ stitutions which transform into itself the domain R. By virtue of (3), any substitution of the group G does not change the form oj . The group g being adjoint to the group G, one con¬ cludes that each substitution of the group g will only permute te linear forms (4) by changing the sign of a few among them. By noticing that x\ + x\ + . . . + x 2 n + (®1 + x 2 + ■ ■ ■ + X n ) 2 = 2 ip, one concludes that the group g is composed of all the substitutions which permute the forms x\ + x\ + . . . + x 2 n + (xi + x 2 + ■ ■ ■ + X n ) 2 . Let us indicate xq = —x\ — X 2 — ■ ■ ■ — x n and x' 0 = — x\ — x' 2 — ■ ■ ■ — x' n , (5) and let ko,ki,...,k n be any one permutation of numbers 0, 1, 2, . . ., n. By posing Xi = eix' k . where e, = ±1, (i = 0, 1,2, . . ., n) (6) one will have Xo + X 1 + . . . + x n = e 0 x' ko + eix' kl + . . . + e n x' kn , and as, because of (5), Xo = Xi + . . . + x n = 0 and x' 0 , x[, . . . , x' n = 0, it is necessary that eo — ^ 1 — • • • — therefore the equalities (6) reduce to the one here: Xi = ex' k .. (i = 0, 1, 2, . . ., n; e = ±1) (7) 70 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series The number of substitutions defined by the formulae obtained is equal to 2-1-2 ■■■ (n +1). By not considering as different two substitutions of opposite coefficients, one will say that the group g is composed of (ni)! different substitutions. § With the help of substitutions (7), one can transform any perfect form (2) contiguous to the principal form into another form contiguous to the form c p, arbitrarily chosen. We have arrived at the following important result. All the perfect forms contiguous to the principal per¬ fect form are equivalent. Let us choose one form among those of the series (2). Let us declare (pi = ip - px !X 2 - All the perfect form contiguous to the form ip are equivalent to the form <p\. Let us find the corresponding value of the parameter P- As we have seen in Number 22, the value searched for of p presents the smallest value of the function Vjxi, X 2 , ■ ■ ■ , X n ) - 1 XiX 2 ( 8 ) determined with condition xix 2 > 0 . One will distinguish in the subsequent studies two cases: 1). n = 2 and 2). n> 3. § See: Minkowski, Zur Theorie der positiven quadratis- chen Formen [On the theory of the positive quadratic forms], (This Journal, V. 101, p. 200) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 71 Monograph Translations Series G. F. Voronoi (1908) First case: By comparing two n = 2 binary forms tp = xf + + X 1 X 2 and ip 1 = Xi X 2 X 1 X 2 — px 1 X 2 , one will notice that by making p = 2 one obtains the form <Pl = x\ + x\ - X1X2 which is evidently equivalent to the perfect form ip, therefore the perfect form ip 1 is that which one has searched for. Second case: By making Xx = 1 , X 2 = 1 , X 3 = - 1 , 034 = 0 , = 0 , one obtains a value of the function (8) which is equal to 1 . By making p= 1, one will present the form cpi under the following form: Fi = ^ [Oi, x 2 , ■ ■ ■ , X n ) 2 + (xx - X2) 2 + xj + . . . + x 2 n \ . (9) It results in that the form p>x is positive. On the ground of that which has been said in Number 23, one will find now all the systems of integers verifying the inequality ipx(xx,x 2 , • • •, X n ) < 1. By noticing that the inequality if l(2:i, X 2 , • • • , X n ) < 1 is impossible, because the positive form ip 1 has integer values which corresponds to the integer values of vari¬ ables, one concludes that the form p>x is perfect. 72 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series With the help of the equality (9), one will easily de¬ termine all the presentations of the minimum of the per¬ fect form ip i. On the binary and ternary perfect forms and on the domains which correspond to them. The binary principal perfect form 3 p = x 2 + xy + y 2 , D = — possesses, as we have seen in Number 29, three contigu¬ ous perfect forms which are equivalent to the principal form. One concludes that all the perfect binary forms con¬ stitute only a single class of forms equivalent to the prin¬ cipal form. The domain 1Z corresponding to the principal form is made up of binary forms (a, b, c) which are determined by the equality ax 2 + 2 bxy + cy 2 = px 2 + p'y 2 + p"(x - y) 2 where p>0, p' > 0, p” > 0 It follows that the domain R is determined by the inequalities p = a + b > 0, p' = — b > 0, p" = c + b > 0. By calling reduced the positive binary forms verify¬ ing these inequalities, as we have done in Number 27, one will establish a well known method of reduction, due to Mr. Selling, f f Selling. Uber die binaren und ternaren quadratis- chen Formen. [On the binary and ternary quadratic forms] (This Journal, V. 77, p. 143) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 73 Monograph Translations Series G. F. Voronoi (1908) It results in that the domain R° is determined by the inequalities p = c — a > 0, p' = a + 2b > 0, p" = — b > 0. The inequalities obtained only differ from famous conditions of reduction of positive binary quadratic fo¬ rms due to Lagrange by the choice of the sign of the coefficient b, that which one can arbitrarily make in the method of Lagrange. | Let us examine now the ternary perfect forms. The principal perfect form ¥> = x 2 + y 2 + z 2 + yz + zx + xy, D = \ possesses six contiguous perfect forms which, all, are equivalent to the perfect form <Pi = x 2 + y 2 + 2 2 + yz + zx which we have found in Number 31. The substitution x = —x', y = y 1 , z = -y' - z' transforms the form y>i into principal form. One concludes that all the ternary perfect forms form only a single class. J See: Lagrange. Recherches d’Arithmetique. [Studies in arithmetic] (Oeuvres de Lagrange published by Serret, V. Ill, p. 698) Gauss. Disquisitiones arithmeticae, art. 171. (Werke, V. I.) Lejeune Dirichlet. Vorlesungen fiber Zahlentheorie [Lec¬ tures on number theory], published by Dedekind, (Brau¬ nschweig 1894, §64, p. 155) 74 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series The domain R corresponding to the principal form is made up of all the ternary quadratic forms (a a' a "\ V b b' b") which are determined by the equality ax 2 -j- a'y 2 -j- a" z 2 -j- 2 byz -j- 2h'zx + 2b" xy = Pi + P 2 V 2 + P 3 Z 2 + P4(y ~ z) 2 + p 5 (z - x) 2 -\- p 6 (x - y) 2 . The domain R is determined by the inequalities pi = a -\- b' -\- b" > 0, p 2 = a' b" -j- b > 0, p 3 = a " b -)- b' > 0, P4 = —b > 0, p 5 = ~b' > 0, p 6 = -b" > 0. By calling reduced the positive ternary quadratic fo¬ rms belonging to the domain R, one will establish a method of reduction due to Mr. Selling. The domain R can be partitioned into 24 equivalent parts which can be transformed one into another with the help of 24 substitution adjoined to those which do not change the principal form. One of these parts, the domain 1Z, will be composed of all the ternary quadratic forms determined by the equality ax 2 + a'y 2 + a"z 2 + 2 byz 2 b'zx + 2 b"xy = pix 2 + p 2 y 2 + P 3 Z 2 + p 4 (y - z ) 2 + p 5 ^t + p [6 ]W where * = * 2 + y 2 + * 2 (y-z) 2 + (z~x) 2 , w = x 2 + y 2 + z 2 {y - z) 2 + (z - x) 2 + (x - y) 2 . God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 75 Monograph Translations Series G. F. Voronoi (1908) One will determine the domain 1Z with the help of inequalities pi = a + 26' + b" > 0, P2 = a' b -j- b' -j- b" 7 > 0, p 3 = a " + b + b' + b" >0, p4. — —b -j - b' ^ 0 , P5 = — b' b" > 0, p 6 = -fe" > o. The domain 1Z enjoys the following properties: 1. Any positive ternary quadratic form is equivalent to at least one form belonging to the domain 1Z. 2. Two ternary quadratic forms which are interior to the domain 1Z can not be equivalent. By effecting the transformation of the domain 1Z with the help of all the substitutions of integer coefficients and of determinant ±1, one will make up the set (7 Z) of domains. Each domain 1Z belonging to the set (7 Z) possesses six domain contiguous by faces in 5 dimensions. The domain 1Z will be transformed into contiguous domain with the help of the following substitutions: 76 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series Each substitution of this series transforms into it¬ self a corresponding face in 5 dimensions of the domain 1Z and permutes two domains of the set (1Z) which are contiguous through this face. This results in a method for the search for the sub¬ stitution which transforms a given form into a form be¬ longing to the domain 1Z. This method is analogous to that which has been shown in Number 27. By calling reduced any positive ternary quadratic form belonging to the domain 1Z, one will establish a new method of reduction of positive ternary quadratic forms which can be considered as a generalisation of the method of reduction of Lagrange. On the perfect form x\ x\x\ + * 1 X 3 + X 1 X 4 + ...-(- X n — lX n . Let us examine the perfect form <Pi = + x\ + . . . + x 2 n + xxx 3 + xix 4 + . . . + x n -ix n obtained in Number 31. One has admitted Qjii — 1 j (i — ^12 — 0 5 ^ • (z = 1 , 2 , ... ,n-,j = 1,2 ,...,n\ij= 3 ) It results in that God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 77 Monograph Translations Series G. F. Voronoi (1908) By supposing that n > 4, one will have n 2 — n repre¬ sentations of the minimum of the form ip\ the number of which is greater than n ( n +D . These representations of the minimum of the form ip i will be characterised by the linear forms A i — x i, A 2 — x 2 , ■ ■ ■ , A n — x n , An + l — X± &3, • • • 5 A w(w+1) , - X n —\ Xfly A Tt(Tl-t-l) - X± X2 3^35 * * * 5 , , ( 1 ) A „(„ + 1 ) | n 3 = XI + X 2 - x n , v ' A rt(rt-fl) j ^ ^ — Xi -(- X2 3^3 X4, ■ ■ ■ ) . A-n, 2 —ro — “(“ X 2 X n — i 'C-n,* The domain f?i corresponding to the perfect form ip\ is made up of forms determined by the equality n 2 — n f(x 1, x 2 , ■ ■ ■ , X n ) = Pk^l where p k > 0. k = 1 (k = 1,2, . . . ,[n 2 - n]) Let us find the linear inequalities which define the domain R\. The number of these inequalities is so large in deed for n = 4. One will overcome the difficulties which result by the help of a particular method. Let us find the group g i of substitutions which do not change the form ip\. To this effect, let us introduce in our studies a quad¬ ratic form u determined by the equality ^ _ 2 (Ai + A2 + • • • + A^ 2 _ n ). 78 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series After the reductions, one obtains cc( xi, x 2 , ■ ■ ■ , x n ) =nx\ + nx\ + 4^3 + . . . + 4x 2 n + 2(n - 2)xix 2 - 4xxx 3 - . . . - 4xix n - 4x 2 x 3 - ... - 4x 2 x n . One can give in the form oj 2 the following expression: u(xi, X 2 , . . . , X n ) =(X! - x 2 ) 2 + (xi + x 2 ) 2 + ( X 1 + x 2 - 2x 3 ) 2 + • • • + (X! + x 2 - 2x n ) 2 . It is easy to demonstrate that the form added to the perfect form ipi has coefficients which are proportional to those of the form 00 . It follows that the perfect form gp 1 is extreme. Let us observe that the linear form Xl + X 2 + . . . + X n , X 1 - X 2 , X 3 , X 4 , ■ ■ ■ , X n characterise n minimum 4 representations of the form 00 . In the case n > 5, other representations of the minimum of the form u do not exist; in the case n = 4, one obtains 12 representations of the minimum of the form c 0 . By noticing that 1 = ^ [(^1 + X 2 + . . . + X n ) 2 + (Xx - X 2 ) 2 + x\ + . . . + X 2 ] , one can say that the group g±, in the case n > 5, is com¬ posed of all the permutations of the forms (xi + X 2 + . . . + Xn) 2 , ( Xi-X 2 ) 2 , xl,..., x 2 n . In the case n = 4, one will determine by this method only divisor of the group g±. By indicating Ml = Xx + X 2 + . . . + X n , God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 79 Monograph Translations Series G. F. Voronoi (1908) u2 = Xi - Xz, U3 — • • • 5 ^n — > u) = x\ + x ' 2 + ... + x' n , u 2 = X 1 — X 2 , u ' 3 = x' 3 , . . . ,u' n = x' n let us declare Ui = eiUf.., (i = 1, 2,. . ., n) (2) where = ±1 (z = 1,2, . . ., n) and the indices Aq, k 2 , ■ ■ ■ , k n present any one permutation of numbers l,2,...,n. Each system of equalities (2) defines a substitution of the group g\. One concludes that the group g± is composed of - 2 n ~ 1 n! different substitutions, in the case n > 5. Let us suppose that the domain R 1 be determined by the inequalities ^2Pij )a i > °- (fc = 1 , 2 , . . . , ct) By indicating ^k(x i,x 2 , ■ ■ ■ ,x n ) = J2pffx*Xj, = 1 , 2 , • • •, a) one will determine, as we have seen in Number 22, a perfect forms ¥>[ k) = Vi + Pk^k (k = 1,2, . . . , a) (3) contiguous to the perfect form ipi. All the substitutions of the group g± will make only one permutation of forms 'Ll, ^ 2 , ■ ■ ■ , (4) 80 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series Let us effect the transformation of forms (3) and (4) with the help of the substitution Xi,x 2 .,x„ = x\, X! - x 2 = x' 2 , x 3 = x' 3 , ... , x n = x' n . (5) The series (4) will be transformed into a series * 1 , * 2 , ■■■,*'«,■ Let us indicate by g a group of substitutions xi 5 (i — 1 ) 2 ,..., n ) where e = ±1 (i = 1 , 2 , . . ., n) and Aq, & 2 , ■ ■ ., k n present a permutation of numbers l,2,...,n. Each substitution of the group g makes only one permutation of forms ( 6 ), and to a similar substitution corresponds a substitution of the group g\. By indicating V k (x i, a= 2 , • • •, x' n ) = ^P^x'iXj, (fc = 1 , 2 ,. . ., a) one will determine with the help of inequalities J^P^aiCLjy 0, (fc=l,2,...,a) (7) a domain 1Z. The form ip\ will be transformed into a form 2 \ X 1 + X 2 + ■ ■ ■ + X n ), with the help of the substitution (5), and any system (xi,X 2 , . . . , x n ) of integers xi,X 2 ,---,x n will be replaced by a system (x^, x' 2 , • • • , x ' n ) of number, also integer, x' x , x 2 , . . . , x' n satisfying the condition x[ + X 2 + . . . + x' n = 0(mod 2). ( 8 ) It results in that the linear forms (1) which corre¬ spond to the various representations of the minimum of the form <pi will be replaced by the forms x'i + x) and x\ - x) (i = 1 , 2 , . . . , n; j = 1 , 2 , . . ., n; i ^ j) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 81 Monograph Translations Series G. F. Voronoi (1908) which characterise the various representations of the m- inimum 2 of the quadratic form x' 2 + x' 2 2 + . . . + x' n 2 , in the set ( X ') of all the systems (x ' x , x' 2 , ■ ■ ■, x' n ) of integers x \, x ' 2 , • ■ ■ , x' n satisfying the condition (8). One concludes that the edges of the domain 1Z will be characterised by the quadratic form (xl + x)) 2 and (x\ - x)) 2 . (i = 1,2 , ,n; j = 1,2 ,. . ., n; i j) By virtue of (7), one obtains the inequalities + 2Pj k) + P {k) > 0 and P} z k) - 2 pj k } + pj k) > 0. (fe = l,2,...,CT;z=l,2,...,n;j = l, 2 ,...,n;Z 7 ^j) ( 9 ) Let us examine any one form ^'(x'i,x 2 , . . . , x' n ) = Pijx'iX j (10) belonging to the series (6). By virtue of (9), one will have Pa 2 Pi j Pjj > 0 and Pa — 2 Pij -I - Pjj ^ 0. (i = l,2,...,n;j = l,2,...,n;i^j) (11) Among these conditions one will find t quantities which define the coefficients of the form (10) to an im¬ mediate common factor. All these equalities will be of the form Pkk 2CkhPkh Phh — 0 where Ckh — il. (12) Let us suppose that there exists a combination of values of k and h satisfying the conditions Pkk + 2 Pkh + Phh > 0 and Pkk ~ 2Pkh + Phh > 0. (13) 82 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series By noticing that the coefficient Pkh does not enter the other inequalities (11), one concludes that the coef¬ ficient Pkh remains undetermined. For all the coefficients of the form (10) to be deter¬ mined by the conditions (12) to an immediate common factor, it is necessary, the coefficient Pkh being indepen¬ dent of other coefficients, that all the coefficients which remain cancel out. By virtue of inequalities (13), this supposition is im¬ possible, therefore the inequality (12) has to hold for all the values of indices k and h. One obtains n ^ n ~ 1 '> conditions P k k ~ 2 e k hPkh + Phh = 0 where e k h = ±1. (fe = l,2,...,n;h=l,2,...,n;fe^=h) (14) which serve to determine the coefficients P k h in functions of coefficients Piii P 22 , ■ ■ ■ , Pnn- (15) The coefficients P 11 , P 22 , ■ ■ ■ , Pnn can not be indepen¬ dent, and will be connected by at least n— 1 equations of the form (12). Therefore, in at least n— 1 case, one will have the equations of the form Pkk ± 2P k h + Phh = 0. (16) To make short we will call these equations double. This stated, let us suppose, in the first place, that there exists at least one coefficient among those of the series (15) which does not enter in the double equations (16). One can suppose, to fix the ideas, that Pn be such a coefficient. The coefficient Pn being independent, all the coeffi¬ cients P22, ■ ■ ■ , P nn will cancel each other and, by virtue of (14), the coefficients B23, -P24, • • • , P n -l,n God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 83 Monograph Translations Series G. F. Voronoi (1908) will also cancel one another out. The coefficient P±i is used for determining the coef¬ ficients P 12 , P13, ■ ■ ■ , Pin with the help of equations (14) which take the form Pn — ZeikPik = 0; (k = 2, 3, . . ., n) it follows that 2Pik = eifePn. (k = 2, 3n) As, on the ground of the supposition made, Pn + 2eikPik > 0, (fc = 2, 3, . . . , n) it is necessary that P li > 0, and one can declare P ii = 1. The form (10) is determined by the equalities ob¬ tained, and one will have (x I , X 2 , i) — x'i + ei2X l 1 x' 2 + • • • + ei r x[ x'n (17) By replacing the variables 612^2, ^13X3, • • • , & i n X n by the variables x' 2 , ■ ■ ■ , x' n , one will replace the form (17) by the form (x h X 2 i • • • : X 7l ) — X1 (x 1, X 2 j • • • , X n ) . Let us suppose, in the second place, that all the co¬ efficients (15) enter in the double equations (16). 84 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series At least one of the coefficients (15) is not zero. Let us suppose that Pkk 0. Following the hypothesis, the coefficient Pkk enters in at least one double equation P k k±2P kh + P hh = 0. It follows that l’kh — 0 and Pkk Phh — 0) therefore the coefficients Pkk and P^h are of opposite signs. Let us suppose, to fix the ideas, that Pn = -1. (18) By examining the inequalities Pi i ± 2 Pi k Pkk > 0, (fc = 2,3,...,n) one deduces Pkk > 0. (k = 2,3, . . . ,n) It results in that the double equation Pkk ± 2 P k h + Phh = 0 has to be impossible so long as k > 2 and h > 2, therefore all the double equations will be of the form Pii±2Pi k + P kk = 0. (fc = 2,3, ...,n) From these equations one gets, by virtue of (18), Pkk = 1 and Pik = 0 . (fc = 2, 3, . . ., n) (19) By substituting the values obtained of coefficients Ph,P 2 2,..., Pnn in the equations Pkk ~ 2e kh Pkh + Phh = 0 where e kh = ±1, God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 85 Monograph Translations Series G. F. Voronoi (1908) one obtains, because of (19), F*kh —— &kh where e^kh —— il- (k = 2,3, ..., n; h = 2,3, ..., n; k h) The form (10) will have for expression ^\ x 'i,x' 2 , . . . ,x' n ) = -x' 2 + x' 2 2 + x' 3 2 + . . . + + 2e23x' 2 x' 3 + 2e24®2a;4 + . . . ^ e n-l,n x n -l X n where e 2 3 = ±1) e 2 4 = ±1, . . ., e n -i }n = ±1. ( 20 ) „ _ (n — l)(n — 2 ) , . „ _ One obtains in this way 2 2 different forms. By permuting the variables and by changing their signs, one will particularly decrease the number of various forms determined by the formula (20). With the help of results obtained, one can easily recognise whether a given quadratic form J2 a ij x i x j be¬ longs to the domain 1Z or not. One will examine, in the first place, the sums eifeOife + e 2 fca 2 fc + ...-)- e n kd n k where e lk = ±1, e 2fc = ±1, . . . , e nk = ±1 and e k k = 1- (k = 1,2,..., n) All these sums have to be positive or zero. The in¬ equalities ®kk |0Hfc| • • • |^fc — l,fc| 1 ,/c | • • • l^ra/cj ^ 0, (fc=l,2,...,n) (21) present the conditions necessary and sufficient for the inequalities eifcOtift -(- €-2k® J 2k ^-nk^nk ^ 0 86 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series (k = 1,2, . . . ,n) to hold. Let us examine, in the second place, the inequalities — a 11 + <*22 + 033 + • • • + a nn + 2e23 a 23 + 2e24®24 +••• + 2e n _i jn a n _i ;n > 0 where e 2 3 = ±1, e 2 4 = ±1, • • •, e n _i >n = ±1, These inequalities can be replaced by a single one “On + O 22 + O 33 + . . . + a nn — 2 | 023 | “ 2|a24| “ • • • “ 2 | <3. ^— 1,7T. | > 0- One will present this inequality under the form on + a 22 + • • • + a nn 2 |a 12 > 2 (an By permuting the variables, one obtains n inequali¬ ties On + O 22 + • • • + o nn — 2 |ai 2 | — 2|ai3| — ... — 2|a n _i >n | > 2 (afcfc — | a i fc | — . . . — |a n fc|) where k = 1,2 , ,n (22) We have arrived at the following result. One can easily recognise whether a given positive quadratic form / belong to the domain f?i or not. To this effect, one will transform the form / by a form /' with the help of the substitution adjoint to the substitution (5) and one will examine 2n inequalities (21) and (22). For the form / to belong to the domain R i, it is necessary and sufficient that the form /' verifies 2 n inequalities (21) and (22). 38 Let us return now to the perfect forms (3) contiguous to the perfect form tp±. We have seen that these forms • • • 2|a n _i jn ■ la. 12 1 - ... - lain I) • God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 87 Monograph Translations Series G. F. Voronoi (1908) will be transformed with the help of the substitution (5) into forms \ [x'x 2 + x ' 2 2 + ... + x' n 2 ) + Pk^' k {x i, ai'a, • • • ,x' n ). (k= 1,2,..., a) The forms T' 2 , . . . , T' CT can be transformed with the help of substitutions belonging to the group g into forms ' !)• x'zi-x'x - x' 2 + x' 3 + x' 4 + . . . + x' n ), - x' 2 2 + x\ 2 + x' 3 + . . . + x' n 2 - 2 x' 1 x' 3 - . . . . 2XiX n “1“ 2634 X 3 X 4 “j" • ■ • 2 e n -i jn X n _ 1 X n , (23) where C 34 Tlj ■ ■ ■ ■ €-71 — 1,71 Tl. The inverse substitution to the substitution (5): x'l = Xi + X 2 + • • • + x n , X 2 — X1 X 2 , X3 x 3 ,, x' n = x n will transform the forms (23) into forms 1 ) . - 2 xix 3 2 ) . 4 (xix 2 - S 3 4X 3 x 4 - ... - ix n ), where; 634 = 0 or 1 , ... , 6 n -i t7l = 0 or 1 . One concludes that all the perfect forms contiguous to the form cpi are equivalent to the following perfect forms 1) -¥>i - PX 1 X 3 , 2) .</?i d" p(xix 2 634X3X4 ... 6 n —i, n x n — 1 x n ), where 634 = 0 or 1, . . . , 6 n - i, n = 0 or 1. Study of the perfect form y>i — px 1X3. 88 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series The perfect form ip i, possesses, as we have seen in Number 38, many contiguous perfect forms which are not equivalent. One will determine in the following only a single per¬ fect form ¥>2 = <Pi ~ px±x 3 contiguous to the perfect form yq. We have demonstrated in Number 22 that the pa¬ rameter p presents the smallest value of the function Oi, x 2 , ■ ■ ■ , x n ) - 1 P = Fi -——- ( 1 ) X i X 3 determined on condition that x±x 3 > 0. (2) By declaring 33 1 = 1, X 2 = 0, a3 3 = l, 33 4 = -l, 335 = 0,..., X n = 0, one obtains the value of the function (1) which is equal to 1, therefore 0<p<l. (3) Let us effect the transformation of the function (1) with the help of the substitution 33 3 = x 'l 1 — 33 1 + 33 2 = x‘ 2 , Xl + 33 2 + • • • + X n = -33'g, X4 = X4 , . . . , x n = x n one will have = 33' 1 2 + 33' 2 2 + .. , + 33' n 2 -2 P -X’ x ( 33 [ + X' 2 + . . . + X’ n ) where, because of (2), Xi(x '1 + x' 2 + . . . + x ' n ) <0 (4) (5) ( 6 ) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 89 Monograph Translations Series G. F. Voronoi (1908) and, because of (4), x\ + x ' 2 + . . . + x' n = O(mod 2), the variables x\ + x ' 2 + . . . + x' n being integers. Let us indicate f(x 1 ,x 2 ,---iX n )=xl+xl + ...+Xn+px 1 (x 1 + X2 + ... + x n ). By virtue of (5) and (6) the value looked for of p is defined by the conditions that the inequality f(xi,x 2 , • • • , x n ) < 2 is impossible, so long as the integers x±, x 2 , ■ ■ ■ , x n verify the congruence Xi + X 2 + . . . + x n = O(mod 2), (7) and that there exists at least one system (Zi, l 2 , . . ., l n ) verifying the equation f(xi,x 2 , • • •, x n ) = 2 (8) and the congruence (7). The form / can be determined by the equality ( gp 1 \ 2 / + {X 3 + P — J + + [x n + p^-j +^l + p- n — 1 P x ± . ( 9 ) It follows that the form / will be positive, provided that 1 + P n — 1 -p 2 > o, and the upper limit R of values of p verifies the equation n — 1 1 + R - R 1, 90 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) therefore Monograph Translations Series R = 2 \fn - 1. ( 10 ) This presented, let us examine a system (l i, l 2 , . . . , l n ) of integers verifying the equation (8) and the congruence (7). I say that there will be the inequalities ; jl h il+p 2 < 1. (i = 2, . . . ,n) ( 11 ) In effect, if one suppose that < 1, Ik + P~2 one will determine e k = ±1 such that the inequality Ik + 2 ek + — < Ik + P~2 holds, and one will present l'i = li and l' k = l k = 2efc. (i = 1 , 2 , . . . , n; i k) The condition (7) will be satisfied, and one will have, by virtue of (9), the inequality /(z'i,r 2 ,...,o <2, which is contrary to the hypothesis. By examining the inequalities (11) and the form / with the help of the formula (9), one will easily demon¬ strate that among the system of integers verifying the equation (8) with condition (7) is found at least one sys¬ tem (Zi, 1 2 , . . . , l n ) satisfying the conditions f(h,h,...,ln) = 2 (12) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 91 Monograph Translations Series G. F. Voronoi (1908) and l 2 = l 3 + 5, I 3 = I 4 = . . . = l n where (5 = 0 or ±1. (13) By virtue of ( 6 ), one will have the inequality £1 [^i + <5 + (n — 2 ) 13 ] < 0 . One can suppose that h<0, (14) and it follows that £i + <5+(ti — 2)^3 >0, therefore, because of (13) and (14), it is necessary that h > 0 . I say that 1 3 = 1. To demonstrate this, let us ef¬ fect the transformation of the positive quadratic form f (xi, X 2 , ■ ■ ■, x n ) with the help of the substitution xi = -x, x 2 = y, x 3 = x 4 = • • • = x n - z-, (15) one will obtain a ternary positive form F(x, y, z) = x 2 + y 2 + (n - 2)z 2 - px(-x + y + (n - 2)z). By virtue of the condition (7), the integers x,y,z ver¬ ify the congruence x + y + (n — 2 )z = 0 (mod 2 ). (16) By indicating u = —l 1 , v = 1 2 , w = l 3 , one will have, because of (12), (13) and (15), F(u, v,w) = 2, 92 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series and the condition (16) will be fulfilled. The inequality f( x ,y,z ) < 2 is impossible so long as the integers x,y,z verify the con¬ gruence (16). Let us effect the transformation of the form F(x,y,z ) with the help of the substitution x = x, + y'+ (n - 2)z', y = x' - y 1 , z = z 1 . (17) The set of systems ( x,y,z ) of integers verifying the congruence (16) will be replaced by the set of systems ( x',y',z ') of arbitrary integers. Let us indicate by F 1 (x', y ', z 1 ) the transformed form. Let D and D' be the determinants of forms F(x,y,z ) and F'(x', y', z'). By virtue of (17), one will have D' = 4D. (18) Let us notice that the number 2 presents the mini¬ mum of the form obtained F'(x 1 , y ', z 1 ) determined in the set of all the systems (x 1 , y' , z ') of integers, the system (0,0,0) being excluded. On the ground of the known theorem § on the limit of the minimum of a ternary positive quadratic form, one will have the inequality 2 < yfl2D 7 . § See: Gauss. Werke, V. II, p. 192, Gottingen 1863. Lejeune-Dirichlet. Uber die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. (This Journal, V. 40, p. 209) Hermite. Sur la theorie des formes quadratiques ter- naires. [On the theory of ternary quadratic forms] (This Journal, V. 40, p. 173) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 93 Monograph Translations Series G. F. Voronoi (1908) It follows that D' > 4, and because of (18), one obtains D >1. (19) This presented, let us observe that the form F(x, y, z ) has the following values: F(u,v,w) = 2, F( 1,1,0) = 2, F(l,-l,0) = 2 + 2p. By transforming the form F(x, y, z) with the help of the substitution (u, 1 , 1 \ v, 1, -1 , (20) \w, 0, 0 / one obtains a form F 0 (x', y', z') ax J 2 + a'y' 2 + a"z rz + 2 by'z' + 2 b'z'x' + 2b"x'y', ^ // _/2 where o = 2, a 1 = 2, a" = 2 + 2 p and b = p. (21) The product a a' a" in any positive ternary quadratic form ^ ^ is, as one knows, always greater than the determinant of the form, unless the coefficients b, b ', b" do not simultaneously cancel one another out. By indicating with Do the determinant of the form Fq(x', y' , z 1 ), one will have, because of (21), D 0 < 4(2 + 2 p), and as, by virtue of (20), D 0 = 4 w 2 D, it becomes w 2 D < 2 + 2 p. 94 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series By virtue of (3) and (19), one obtains the inequality w 2 < 4, therefore w = 1. By returning to the equalities (13), one obtains 1 1 = — u, I2 = S and I3 = 1, I4 = 1, . . . , l n = 1, where u> 0 and 5 = 0,1,2. By substituting the values found of Z 1 , Z 2 , function (5), one will have P = u 2 + S 2 + n — 4 u( — u + 5 + n — 2) , l n in the ( 22 ) It remains to determine the smallest value of this function providing that u > 0, — u S n — 2>0, u = n + 5(mod2) and 5 = 0,1,2. (23) Let us admit u = \fn — 1 + a, a being a real number. The function (22) takes the form _ 2n + (2a —2) v /n + a 2 —2a + 5 2 —3 P ij/n + (a — 2)n + (5 — 2a)y / n + 1 — a 2 + a5 — 5 (24) ity The value searched for of p has to verify the inequal- P < R, God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 95 Monograph Translations Series G. F. Voronoi (1908) therefore because of (10), one will have 2 - 1 - p > 0. (25) After the reductions, one obtains 2 s/n- 1 ~ P ~ (1 — <5 2 — (- 2 <5 — a 2 )\fn 5 2 — 25 — 1 — a 2 — 2a 2a5 (\fn — 1) [n\fn + (a — 2)n + (5 — 2 a)\/n + 1— a 2 + a5 — (5] and, because of (25), it becomes (1 - 5 2 + 25 - a 2 )^/n + 5 2 - 25 - 1 - a 2 - 2a + 2a5 > 0. By noticing that 5 2 - 25 - 1 - a 2 - 2a + 2a5 <0 so long as <5 = 0, 1,2, one obtains the inequality 1 - 5 2 + 25 - a 2 >0. By making (5 = 0 and 2, one will have a 2 < 1 as long as (5 = 0 and 2 (26) By making 5=1, one will have a 2 < 2 so long as 5=1. (27) Let us indicate by m a positive integer determined with the help of inequalities \fn — 1 < m < \fn. (28) By declaring n = m 2 + p, (29) 96 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series one will have a positive integer p verifying the inequali¬ ties 0 < p < 2m + 1. (30) First case: p is an odd number. By virtue of (23) and (29), one will have a congru¬ ence uErn 2 -|-p + i5(mod2), p being an odd number; one can declare u = m 2 + 5 + 1 + 2t. (31) By declaring Vn = m + £, one will have 0<€<1, (32) because of (28). By virtue of (24), one obtains the equal¬ ity u = m — 1 + £ + a, and because of (31), it becomes £ cx =■ m ^ — m —[— 2 —(— 2t -(- 5. (33) By supposing that (5 = 0 or 2, one obtains £ + a = 0(mod 2). By virtue of (26) and (32), it is necessary that 4 + a = 0, therefore u = m — 1 so long as (5 = 0 and 8 = 2. By supposing that 5 = 1, one obtains, because of (33), £ + a = l(mod 2). God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 97 Monograph Translations Series G. F. Voronoi (1908) By virtue of (27) and (32), the integer £ + a can have only two values £ + a = ±1, and it results in that u = m or m — 2 as long as 5=1. One obtains four values of the function (22): (to — l) 2 + ra — 4 _ TO 2 + n —3 Pi P3 (m—l)(n —m—1) ’ (m — 2) 2 -|-n — 3 P2 P4 = m(n — m — 1 ) ’ (m — 2) 2 + n (m—2) (n —m-f-1) 5 ( m — l)(n — ra + 1) among which is found the smallest value looked for of p. By noticing that _ _ Pj2_ 3_ P 1 P 2 m(m — l)(n — m — 1) ’ _ _ 2p+ 2 _ ^ 1 (m — l)(n — m — l)(n — m + 1) ’ _ P + 1 _ (m — l)(m — 2)(n — m + 1) ’ one obtains, because of (30), Pi < P4 < P3 and p 2 < Pi so long as p > 3, Pi < P 2 so long as p < 3. There exists only a single odd value of p verifying the inequalities 0 < p < 3, therefore one will have the inequality Pi < p 2 so long as p = 1. We have arrived at the following result. The smallest value of p will have for expression m 2 + n — 3 m(n — m — 1), (34) 98 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series provided that n = m 2 + p, and the odd number p verifies the inequalities 3 < p < 2m + 1. In the case n = m 2 + 1, the smallest value of p will be _ (m — l) 2 + n — 4 P (m — 1) (n — m — 1) Second case: p is an even number. One will have, because of (23), the inequality [sic] u = m 2 + <5(mod 2). By presenting u = m 2 + (5 + 2 1, one will have the equalities u = m — l+£ + a and £ + a = m 2 — m -\- 2t f- 8 1. By supposing that (5 = 0 or 2, one obtains and it follows that € + a — 11 u = m so long as <5 = 0 and 2. By supposing that (5=1, one obtains £ + a = 0 or £ + a = 2, therefore u = m 1 or u = m -\- 1 so long as (5=1. The smallest value of p is found among the following values of the function (22): _ m 2 + n — 4 hi m{n — m — 2) ’ _ (m —l) 2 -t~n —3 P3 (m — !)(«. — m) ’ P2 P4 fm + l) 2 + 7(. —3 (m + l)(n —m —2) 1 _ m 2 + n m(n — m ) God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 99 Monograph Translations Series G. F. Voronoi (1908) By noticing that P 2 - Pi Pi ~ p4 P4 ~ Pz 2m + 4 — p m(m + 1) (n — m — 2) ’ 4m — 2p m(n — m)(n — m — 2)’ 2m — p m(m — 1) (n — m) ’ one obtains, because of (30), Pz < P4 < Pi < p 2 • We have arrived at the following result: The smallest value of p is expressed by the equality (m— l) 2 + n — 3 P = ---- (m — 1) (n — m) provided that n = m 2 + p, and the even number p verifies the inequalities 0 < p < 2m + 1. We have determined the value of the parameter p which defines the perfect form <pi + px 1X3. The determi¬ nant D of this form, by virtue of (4) and (9), will have for expression D 4 -)- 4p — (n — 1 )p 2 (35) The corresponding value of the function M(aij) de¬ fined in Number 16 will be M(aij ) Y 4 + 4p - (n - l)p 2 ' By applying the formulae obtained to the case: n = 4, 5, 6, 7, 8, 100 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series one obtains the same value of p P = 1 - The corresponding perfect forms will be [ 1 —I | lO [r-1 | ^ [r-H | CO [tH | CN lO c© CN CN CN CN 3 3 1-1 II «l« q II II II II «Z Q Q Q Q I C*3 to CO 1> + 8 H H CO ID CD 8 H H + + + + + 8 + + + + 8 ■S' 4- 8 8 8 8 i oq oo 8 8 8 8 8 + + + + (M oq ID Oq CD oq c~ 8 8 8 8 + <N <M H + + + + + <N <M oq oq oq <m oq oq <N tH 8 8 8 H + + + + <M iH Oq TH oq th oq th 8 8 8 One comes across all these perfect forms in the Mem- oire of Mr.’s Korkine and Zolotareff: Sur les formes quadratiques. [On the quadratic forms] J £ Mathematische Annalen, V. VI, p. 367. God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 101 Monograph Translations Series G. F. Voronoi (1908) The formulae obtained give a mean for the study of various perfect forms which verify the inequality M(aij) > 2 . By making, for example, n = 12, one will have m = 3 and p = 3. By virtue of (34), one obtains 3 P= 4’ therefore, because of (35), 1 212 > and it follows that 12 [16 = 2 y— > 2. All the extreme forms studied by Mr.’s Korkine and Zolotareff do not give a function M(aij) of values which exceed 2. On the quadratic perfect forms and on the domains which correspond to them. We have seen in Number 29 that to the quaternary principal perfect form if = x l + xl + xl + xl -\- XiX 2 -\- XiX 3 -\- XiX^f- X 2 X 3 -\- X 2 X4-\- X3X4, r> — 5 corresponds the domain R made up of forms pixj + p 2 xl + p 3 x% + p 4 xl + P5(xi - x 2 ) 2 + P[ 6] (xi - a3 3 ) 2 + 102 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series P 7 (X 1 -X4) 2 + P8(X2-X 3 ) 2 + P9(X2-X4) 2 + P10(X 3 -X4) 2 ■ All the perfect forms contiguous to the principal form <p are equivalent to the form <Pi = x 1 + x\ + x\ + x\ X±X 3 X1X4 X2X 3 X2X4 -)- X 3 X4, D = b The corresponding domain f?i is made up of forms Pix\ + P2xj + p 3 X% + P4X4 + p 5 (Xi - X 3 ) 2 + p 6 (x 1 - X 4 ) 2 + pr(x 2 - x 3 ) 2 + p 8 (x 2 - X4) 2 + p 9 ( x 3 - x 4 ) 2 + Plo(xi + X 2 - x 3 ) 2 + Pn(xi + X2 - X 4 ) 2 + p 12 ( 2:1 + X 2 - x 3 - X4) 2 . Let us examine the perfect forms contiguous to the perfect form ip\. We have demonstrated in Number 38 that all these forms are equivalent to the forms 1) . - pxix 3 , 2) - + p(x\X 2 — 5x 3 X4), where (5 = 0 or 1. Let us examine three perfect forms !)• 'Pi + px\X 2 , 2). ipx-px 1 X 3 , 3). pi + p(xix 2 - x 3 x 4 )- 1) . By making p = 1 in the form + px 1 X 2 , one obtains the principal perfect form ip. 2) . Let us notice that the form p\— px\x 3 is equivalent to the form p>\ + px\X 2 - In effect, the substitution X\ = —x'x, X2 = x' 3 , X 3 = x' 2 , X4 = x'x + x'4 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 103 Monograph Translations Series G. F. Voronoi (1908) does not change the form tp± and transforms the form X 1 X 2 into the form —x^x^. 3). By making p = 1 in the form tp± + p(x 1 X 2 — X3X4 ), one obtains the form x\ + x\, 33§ + x\ + X\X2 + X1X3 + 331 X 4 + X 2 X 3 + X 2 X 4 which is evidently equivalent to the perfect form yq. One concludes that all the perfect forms contiguous to the perfect form <pi are equivalent to the forms and Wi¬ lt follows that the set of all the quaternary perfect forms be divided into two classes represented by the per¬ fect forms <p and <p ±. The set (f?) of domains corresponding to various qua¬ ternary perfect forms is made up of two classes, too, represented by the domains R and f?i. On the perfect forms in five variables and on the do¬ mains which correspond to them. We have determined two perfect forms in five vari¬ ables if = x\ + xl + . . . + 33 § + 331332 + 33 i 33 3 + . . . + X 4 X 5 , D = ^, Wl = X 1 + x 2 + ■ • ■ + x 5 + X 1 X 3 + X 1 X 4 + • • • + 334335 , D = The corresponding domains R and R± will be com¬ posed of forms R) Pix\ + p 2 xl + . . . + p 5 xl + p 6 (x 1 - 33 2 ) 2 + P7(X 1 - X 3 ) 2 + . . . + P 15 (£4 - 335 ) 2 , Rl) Pix\ + p 2 xl + . . . + p 5 xl + p 6 (xi - X 3) 2 + . . . + P2o(a3l +332-334- X 5 ) 2 . 104 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series Examine the perfect forms contiguous to the perfect face ipi. We have demonstrated in Number 38 that all these forms are equivalent to the forms 1 ) . <pi-px 1 x 3 , ^ 2) . ipi — p(x±X2 — 6x3X4 — S' X3X5 — 5" X4X5). where (5 = 0 or 1 , 8' = 0 or 1 , 5" = 0 or 1 . In the second case one obtains 8 perfect forms. By permuting the variables x 3 , X4, X5 one will replace the forms (1) by 4 forms; thus all the perfect forms con¬ tiguous to the perfect form <pi are equivalent to the 5 following forms: 1. pi + px iX 2 , 2 . ipi + px1X3, 3. ipi + p(xix 2 - X 4 X 5 ), 4 ■ ipi + p(xix 2 - X3X5 - X4X5) , 5. ipi + p(x 1 x 2 - X3X4 - x 3 x 5 - x 4 x 5 ). 1) . By making p = 1 in the perfect form ip 1 + px±x 2 , one obtains the perfect form p. 2) . We have seen in Number 42 that the perfect form Pi — px\x 3 is determined by the value p = 1 of the param¬ eter p in the case n = 5 . One obtains the form Pi = x\ + x\ + x\ + x\ + x\ + X 1 X 4 + xix 5 + • • • + 2 : 42:5 (2) which will be transformed with the help of the substitu¬ tion XI = —X2, x 2 = x\ — x' 2 , X3 = x' 3 , X4 = X 2 + x' 4 , X5 = x' 2 + x' 5 into a perfect form pi. God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 105 Monograph Translations Series G. F. Voronoi (1908) 3 ) . In the form <pi + p(x 1X2 — X4X5 ), one will put p = 0 and one will obtain the form x\ + x\ + . . . + X% + X1X2 + X\X 3 + . . . + X 3 X 5 which is evidently equivalent to the form <pi. 4 ) . In the form <pi + p(xix 3 — x 3 x 3 — X4X5) ), one will put p = 1 and one will obtain the form x\ + x\ + . . . + x 2 5 + XxX2 + XxX 3 + . . . + X2X5 which is evident to the perfect form (2) 5 ) . It remains only to determine the perfect form: T>1 + p(xix 2 - X 3 x 4 - X 3 x 5 - X 4 x 5 ). ( 3 ) By effecting the transformation with the help of the substitution -x 1 + x 2 = x' 1: Xl + x 2 + x 3 + X4 + x 5 = x' 2 , X 3 = x' 3 , X4 = x' 4 , x 5 = x' 5 ( 4 ) of the form 2<pi + 2p(xix 2 - x 3 x 4 - x 3 x 5 - x 4 x 5 ), one obtains the form / 2 1 / 2 1 / 2 1 / 2 1 / 2 I Xl + X 2 + x 3 + x 4 + x 5 + 2 + ^2 + +2:4 +*5 - 2 a 3 2 CC 3 - 2X2X4- 2 x' 2 x' 5 — 2 x' 3 x' 4 — 2 x' 3 x' 5 — 2 x' 4 x' 5 ] . ( 5 ) By virtue of ( 4 ) the integer variables x' 4 , x' 2 , x' 3 , x' 4 , x' 5 verify the congruence x'i + x' 2 + x' 3 + x' 4 + x' 5 = 0(mod2). (6) 106 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series By applying to the form (5) the method unveiled in Number 2, one will determine the value of the upper limit R > 0 of value § with the help of equations Ci — f?€i7 = 0, C2 + 2 - C3 - £4 - C5) = 0 , £3 + R (—£2 + C3 - £4 - Cs) = 0, £4 + f?(—£ 2 — C3 + £4 — Cs) = 0, C5 + R(—£ 2 — C3 — C4 + Cs) = 0 . It results in that C2 = C3 = £4 = £5, and one obtains the equations Ci (1 — R) = 0 and ^ 2(1 — 21?) = 0, thus By declaring cci = 0, x' 2 = 1, x' 3 = 1, x ' 4 = 1, ai'g = 1, (7) one will satisfy the condition ( 6 ) and one will have the value 4 — 4p of the form (5). By making one obtains 4 — 4p = 2, P= 2‘ It follows that the positive quadratic form / 2 . / 2 . . / 2 . + £2 + • • • + X 5 + 4 [-£Ci + a ; 2 + a ; 3 + £4 + x 5 - 2 ir 2 ir 3 - . . . - 2 a 3 4 a ; 5 will have a value 2 corresponding to the system (7). God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 ( 8 ) 107 Monograph Translations Series G. F. Voronoi (1908) By virtue of that which has been discussed in Num¬ ber 23, the smallest value of the form ( 8 ) will correspond to a system (Zi, Z 2 , ■ ■ ■ , Is) verifying the inequality l\ + ^2 + I 3 + I 4 + ^5 < 2 • —- y where R = —. -LX, ^ ^ One obtains the inequality i? + ii + il + /2 + z§<4. It is easy to demonstrate that the system (7) is the only one verifying this inequality on condition ( 6 ), the systems which verify the inequality -aq + aq + 3:3 + 3:4 + a : 5 - 2x 2 x 3 - 2x 2 x' 4: — 2x 2 x' 5 — 2x' 3 x' 4: — 2 x' 3 x' 5 — 2x' 4 x' 5 > 0 being excluded. By making p = \ in the form (3), one obtains the perfect form <Pz = a:i + x 2 + . . . + x ' 5 + -aqaq + aqa : 3 + . . . -\-x2X5 + ^x 3 x 4 + ^x 3 x 5 + ^x 4 x 5 , D 1 25 ' The corresponding domain R 2 is composed of forms Pix\ + p 2 a:i + • • • + P 5 X 5 + Pe(x! - x 3 ) 2 + . . . + pn (x 2 - x 5 ) 2 +P 12 ( 3:1 + a: 2 - a; 3 - x 4 ) 2 + p 13 ( 2:1 + a; 2 - a; 3 - a; 5 ) 2 +p 14 ( 3:1 + x 2 - x 4 ~ X5) 2 + pi 5 ( —aq - a: 2 + a; 3 + 3:4 + a; 5 ) 2 . The number of parameter pi, P 2 , ■ ■ ■ , P 15 being equal to the number of dimensions of the domain R 2 , one will determine without trouble 15 inequalities which define the domain f? 2 . 108 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series We have demonstrated that all the perfect forms con¬ tiguous to the perfect form ip 4 are equivalent to the per¬ fect forms ip,<pi and ip 2 . Choose the perfect forms contiguous to the perfect form ip 2 . To this effect let us notice, in the first place, that the perfect form ip\ is contiguous to the perfect form ip 2 , then observe that all the perfect forms contiguous to the form ip2 are equivalent. To demonstrate this, examine all the faces in 14 di¬ mensions of the domain R 2 . The domain R 2 is characterised by 15 quadratic form- s rp 2 ™ 2 2 ~ 2 1 j **'2 > ^ 3 ’ ^ 4 ’ (xi — X3) 2 , (xi — x 4 ) 2 , (xi - X5) 2 , < (x 2 - X3) 2 , (x 2 - x\), (x 2 - X 5 ) 2 , (9) (xi + X 2 - X 3 - X4) 2 , (xi + x 2 - x 3 - x 5 ) 2 , , (xi + X 2 - X 4 - X5) 2 , (-X! - X 2 + X 3 + X 4 + X 5 ) 2 - Each face in 14 domains of the domain R 2 possesses 14 of these, and the form which remains can be called form opposite to the face. One concludes that each face is well determined by the opposite face. For the perfect forms contiguous to the perfect form ip 2 to be equivalent, it is necessary and sufficient that all the faces of the domain R 2 could be transformed one to one with the help of substitutions which do not change the domain R 2 . It would be easy to write all these substitutions, but one will proceed in another way, more speedy. Let us observe that the face P belonging to the do¬ main R 1 and R 2 is characterised by all the forms (9), the God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 109 Monograph Translations Series G. F. Voronoi (1908) form (— x\ -X2 + X3 + X4 + X5) 2 being excluded. With the aid of substitution associated with the sub¬ stitution ( 4 ), one will replace the forms ( 9 ) by the forms: ' (*; ± x ' 2 ) 2 , (*; ±*') 2 , (*i ±x' 4 ) 2 , ±x' 5 y, < (*' + x') 2 , (x' 2 + x' 4 ) 2 , (x' 2 + x' ) 2 , (x' 3 + x^) 2 , (x' 3 + X[ 5 ]') 2 , . (A + x' 5 ) 2 , (X 2 + x' 3 + x' 4 + x'g) 2 . (10) By changing the sign of x'i and by permuting the vari¬ ables x' 2 , x 3 , x' 4 , x' 5 , one will transform into itself the forms (10), and the form (x' 2 + x 3 + x 4 + A) 2 wi 11 not change. To each similar substitution corresponds a substitu¬ tion which transforms into itself the domain R 2 and the face P of the domain R 2 , and does not change the form (-X1 - x 2 + X 3 + x 4 + X 5 ) 2 . By changing the sign of x’ x and by permuting x 2 , x 3 , x 4 , x' 5 , one will transform the form (x' x + x 2 ) 2 into forms (x'i ± x') 2 , (x' ± x') 2 , (x' ± x' 4 ) 2 , (x' ± x') 2 and one will transform the form (x 2 + x 3 ) 2 into forms (x' 2 + x') 2 ,(x'+x' 4 ) 2 ,(x'+x') 2 , (x' 3 + x 4 ) 2 , (x' 3 + x' [5] ) 2 , (x 4 + A) 2 . Thus only the forms (x'i + x') 2 , (x' 2 + x') 2 , (x' 2 + x' + x' 4 + A) 2 (11) remain to examine. By returning to the forms ( 9 ), one obtains the forms corresponding to the forms (11). x 2, A, (-X1 - x 2 + X 3 + x 4 + x 5 ) 2 . ( 12 ) It is demonstrated that all the forms ( 9 ) can be trans¬ formed into forms (12) with the help of substitutions which do not change the domain R 2 . 110 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. G. F. Voronoi (1908) Monograph Translations Series With the help of substitutions Xi = x ' 2 - x' 5 , X 2 = x' 3 , x 3 = x' 2 , X 4 = x\ + x ' 2 — x ' 4 — x' 5 , X 5 = —x\ + x ' 3 and xi = x\, x 2 = -x\ - x 2 + x ' 3 + x ' 4 + x' 5 , x 3 = x 3 , X 4 — X 4 , x 5 = x' 5 , one will transform the domain R 2 into itself, and the form x 2 will be transformed into forms x 3 and (— x ' 4 — x' 2 -\- x' 3 + A + x ' 5 ) 2 . We have demonstrated that all the forms of the do¬ main R 2 are equivalent. It results in, from that we have seen, that all the perfect forms contiguous to the perfect form ip 2 are equivalent to the perfect form ip One concludes that all the perfect forms in five vari¬ ables constitute three different classes represented by the perfect forms ip, ip i and ip 2 . The set of domains (f?) can be divided into three classes also, represented by the domains R, R\, and R 2 . End of the first Memoire. God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 111